Number Theory
[1]
vixra:2411.0113 [
pdf]
On The Riemann Hypothesis
The aim of this article is to prove that all the roots of the Dirichlet eta function are only on the critical line, which implies that the Riemann hypothesis is true. The roots of the eta and zeta functions in the critical strip are the same. We shall investigate the possibility or not of existence of roots of the eta function which are in the critical strip and not on the critical line.
[2]
vixra:2411.0097 [
pdf]
A Brief Proof that Pi is Irrational
It is shown that the limit of cos(j) and sin(j) as j goes to infinity do not exist. Using DeMoivre's theorem this implies the limits of sin(j!) and cos(j!) don't exist either. Assuming pi is rational, its multiple can be expressed as a factorial. From this a contradiction is derived.
[3]
vixra:2411.0079 [
pdf]
Investigating Fractal Patterns and the Riemann Hypothesis
The Riemann Hypothesis remains one of the most critical unsolved problems in mathematics, proposing that all non-trivial zeros of the Riemann zeta function are located on the critical line. This paper explores unique fractal structures within the zeta function, drawing comparisons with the Mandelbrot set and the Smith chart, which together illuminate possible connections between prime distribution,electromagnetic symmetry, and gravitational principles.
[4]
vixra:2411.0040 [
pdf]
Proving Irrationality of Infinite Series
In this paper a theorem will be proven that is a relatively concise way to confirm the irrationality of an infinite series. From this, new proofs of the irrationality of some known results can be derived(pi, phi) as well as solutions to open problems(rationality of zeta function over the natural numbers, rationality of Euler-Mascheroni Constant)
[5]
vixra:2411.0006 [
pdf]
Preliminary Values for Wild Constant Hunt, Possibly "Nice" Dirichlet Series Originates
This short document purported to publicise assorted constants to be targeted for Reverse Arithmetic (known as unscrambling the egg in pop culture) experimentation. In section one, a tangential intermezzo intended to justify the urgency to archive and some trivia regarding Pie and its defense against the manifesto of otherwise obscure alternative. Later sections detail the constants aimed for Reverse Arithmetic experimentation.
[6]
vixra:2410.0177 [
pdf]
Goldbach Prime Pairs and Its Distribution for Integers of Form 2(n+1)^{2}
In this article, study related to the distribution of the Goldbach prime pairs for the even integers of form 2(n+1)^{2} has been carried out with the help of other conjectures(/postulate), that is, Legendre's conjecture and Bertrand's postulate. From the obtained results, various possible conjectures have been proposed/suggested. In brief, first the term, Goldbach first prime pair (p_{f},p'_{f}) was defined and for ascertaining the interval in which Goldbach first prime pair would always be present for all the values of 2(n+1)^{2}, where, n =1,2,3,..., was suggested. The proposed conjectures on distribution were tested for all values of n = 1,2,3,...,1,000,000 using algorithm. The expression, (n)(n+1) < p_{f} < (n+1)^{2} < p'_{f} < 2(n+1)^{2} - n(n+1) showed that, the Goldbach first prime pair would always existing for all the n values, where, n = 1,2,3,...- {4,6,10,16,19,21,22,23,30,33,36,43,48,49,56, 57,61,66,72, 76,81,106, 117,127,130, 132,141,210,276,289,333,418}. The expression, (n)^{2} < p_{f} < (n+1)^{2} < p'_{f} < 2(n+1)^{2} - (n)^{2} showed that, the Goldbach first prime pairs would always existing for all the n values, where, n = 1,2,3,... - {21,23,30,33,36,48,49,117,141}. While, for the expression, (n)(n-1) < p_{f} < (n+1)^{2} < p'_{f} < 2(n+1)^{2} - (n)(n-1), were true for all the n = 1,2,3,..., values. Another observation was found that, in the proposed conjecture 1, the gap between Goldbach first prime pair, that is, (p'_{f} - p_{f}) was found to be always less than its corresponding n value after n=2538 , hence, Gap(p'_{f} - p_{f}) << n, where, n=2539,2540,2541,....
[7]
vixra:2410.0167 [
pdf]
Proof of the Collatz Conjecture for the Natural Numbers
This document proves that the Collatz Conjecture is true for the Natural numbers excluding zero. Use is made of the probability distribution of even and odd numbers in supposed diverging Collatz sequences to establish that Collatz sequences do not diverge, having a finite number of terms, and are bounded. Finally proof by contradiction, the pigeon hole principle and proof by induction are used to prove that the Collatz Conjecture is true via two theorems.
[8]
vixra:2410.0103 [
pdf]
Any Rational Number N is the Sum and Difference of Four Rational Fourth Powers in an Infinite Number of Ways
It is known that the equation X^4+Y^-(Z^4+W^4) = N has infinitely many rational solutions for any rational number N.We present three new solutions for the equation X^4+Y^4-(Z^4+W^4) = N.
[9]
vixra:2410.0091 [
pdf]
A Simple Proof Of Legendre's Conjecture
We analyze intervals of numbers whose prime factors are all $> y$. In this article we stablish and prove a lower bound of prime numbers contained in the interval $[x^a,(x+1)^a)$ where $a,x in mathbb{R}^+$ and $x>2$, $a geq 2$, then if $a=2$, the interval $[x^2,(x+1)^2)$ contains at least 1 prime.
[10]
vixra:2410.0090 [
pdf]
Fundamental Algebraic Disproof of the Riemann Hypothesis in the Logarithmic Derivative
In this paper we prove the fundamental contradiction about the Riemann Hypothesis, expressing a function as a product and given the following summation, where $R$ is the set of all solutions of $R(x)=0$:begin{equation}frac{R'(x)}{R(x)}=sum_{rin R}left(frac{1}{x-r}ight)end{equation}And considering a regularization for hypertranscendental functions, then the expression applied in Riemann Zeta function of $frac{1}{2}$, or the logarithmic derivative, where $R_t$ is the set of tirivial zeros:begin{equation}frac{zeta'(frac{1}{2})}{zeta(frac{1}{2})}eq sum_{rin R_t}left(frac{1}{frac{1}{2}-r}ight)end{equation}
[11]
vixra:2410.0016 [
pdf]
On the Distribution of the L Function
In this short note, we study the distribution of the $ell$ function, particularly on the primes. We study various elementary properties of the $ell$ function. We relate the $ell$ function to prime gaps, offering a motivation for a further studies of this function.
[12]
vixra:2409.0163 [
pdf]
A New Property of the Sheldon Prime
In this article, the author inquires into a new property for a set of numbers containing the Sheldon prime as the first term that allows to calculate summations just by solving a simple multiplication.
[13]
vixra:2409.0031 [
pdf]
Power-spectral Numbers
Given $n=p_1^{e_1}cdots p_k^{e_k}$, there is a canonical isomorphism $mathbb{Z}_ncongmathbb{Z}_{p_1^{e_1}} oplus cdots oplus mathbb{Z}_{p_k^{e_k}}$. The spectral basis of $mathbb{Z}_n$ is an explicit realization of this isomorphism within $mathbb{Z}_n$ itself. This paper is a study of those numbers whose spectral basis consists of primes and powers. For example, if $M_p$ is a Mersenne prime with exponent $p$, then $2M_p$ has spectral basis ${M_p,2^{p}}$, while $2^pM_p$ has spectral basis ${M_p^2,2^{p}}$. Isospectral and isotropic numbers are introduced and many other numbers with interesting spectral bases are presented.
[14]
vixra:2409.0019 [
pdf]
"Infinitely Often"="infinity" [:] a Statistics Approach to Small Gaps Between Primes
We construct a sequence of consecutive primes. From the perspective of statistics, we analyze and handle them by the combination of the fundamental property of primes with James Maynard's result. It reveals that there are infinitely many pairs of primes which differ by two.
[15]
vixra:2408.0107 [
pdf]
Symmetries in Goldbach's Conjecture
We define a Goldbach table as a table consisting of two rows. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. We then define a novel prime decimal system: it gives for every n remainders when n is divided by all primes less than the n. This suggests linear functions, the divisions used can give another perspective on all the column pairs. The inverses of these functions when put into tabular form give symmetries that suggest Goldbach's conjecture is correct.
[16]
vixra:2408.0080 [
pdf]
A New Method to Find Trivial Zeros of Riemann Hypothesis
The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The counterexamples can be found through this equation.
[17]
vixra:2408.0059 [
pdf]
A Geometric Approach to the Riemann Hypothesis: Analyzing Non-Trivial Zeros in Polar Coordinates
The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line ( Re(s) = frac{1}{2} ) in the complex plane. This paper explores an alternative geometric approach by analyzing the zeta function and its non-trivial zeros in polar coordinates. Transforming the problem into this framework reveals a natural symmetry about the polar axis, which corresponds to the critical line in Cartesian coordinates.We demonstrate that the (Xi(s)) function, a redefined version of the zeta function, retains the symmetry ( Xi(s) = Xi(1 - s) ) in polar coordinates, supporting the hypothesis that non-trivial zeros must lie on the critical line.This geometric perspective suggests a potential simplification in verifying the Riemann Hypothesis and offers new insights into the distribution of non-trivial zeros.
[18]
vixra:2408.0055 [
pdf]
An Advanced Quantum-Resistant Algorithm: Design, Implementation, and Analysis
The advent of quantum computing represents a paradigm shift with profound implications for the field of cryptography. Quantum algorithms, particularly Shor's algorithm, threaten to undermine the security foundations of traditional cryptographic schemes such as RSA, ECC, and DSA, which rely on the computational difficulty of integer factorization and discrete logarithms. As these algorithms become obsolete in the face of quantum capabilities, there is an urgent need for cryptographic systems that can withstand quantum-based attacks. In response to this looming threat, this paper introduces the Quantum Cryptographic Toolkit (QCT), a robust and versatile framework designed to facilitate the development, testing, and deployment of quantum-resistant cryptographic algorithms. The QCT integrates a diverse set of post-quantum cryptographic algorithms, including lattice-based methods like NewHope, code-based approaches exemplified by the McEliece cryptosystem, and isogeny-based cryptography, such as SIKE. Each of these algorithms is implemented with a focus on maintaining security even in the face of quantum computing advancements, addressing both theoretical and practical challenges. The toolkit is structured to be modular and extensible, allowing researchers and developers to seamlessly incorporate additional algorithms and cryptographic primitives as the field evolves. This paper details the design principles underlying the QCT, emphasizing the importance of modularity, extensibility, and performance optimization. We discuss the implementation strategies employed to ensure the toolkit's effectiveness across a range of cryptographic scenarios, from key exchange protocols to encryption and digital signatures. A comprehensive security analysis is provided, highlighting the resistance of each algorithm to quantum attacks, and comparing their performance to other post-quantum cryptographic solutions. In addition to the security analysis, we include extensive performance benchmarks that evaluate the computational efficiency, memory usage, and scalability of the algorithms within the QCT. These benchmarks demonstrate the practical viability of the toolkit for real-world applications, offering insights into the trade-offs between security and performance that are inherent in post-quantum cryptography. The results indicate that the QCT not only meets the stringent security requirements
[19]
vixra:2408.0041 [
pdf]
Geometric Symmetry of Non-Trivial Zeros of the Riemann Zeta Function in Polar Coordinates
This paper investigates the symmetry of non-trivial zeros of the Riemann zeta function (zeta(s)) through geometric analysis in polar coordinates. By transforming the complex number (s = sigma + it) into polar form, we demonstrate that the symmetry about the critical line (sigma = frac{1}{2}) necessitates (sigma = frac{1}{2}) for all non-trivial zeros. Numerical simulations further confirm the accuracy and consistency of this geometric approach. And we introduce a formula for the distribution pattern of all non-trivial zeros:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]where: [r = sqrt{frac{1}{4} + t^2} quad text{and} quad theta = arctan(2t)]
[20]
vixra:2408.0019 [
pdf]
Geometric Analysis of Non-Trivial Zeros of the Riemann Zeta Function and Proof of σ as a Constant
This paper investigates the non-trivial zeros of the Riemann zeta function using polar coordinates. By transforming the complex plane into a polar coordinate system, we provide a geometric perspective on the distribution of non-trivial zeros. We focus on the key formula:[zetaleft(sqrt{frac{1}{4} + t^2} , e^{i arctan(2t)}ight) = 0]This formula reveals the distribution pattern of non-trivial zeros and supports the hypothesis that (sigma) must be (frac{1}{2}) and is a constant.
[21]
vixra:2408.0007 [
pdf]
A Set of Formulas for Prime Numbers
Here I present several formulas and conjectures on prime numbers. I’m interested in studying prime numbers, Euler’s totient function and sum of the divisors of natural numbers.
[22]
vixra:2408.0003 [
pdf]
A Direct Proof that Goldbach's Conjecture is True
We introduce a Goldbach table. It consists of two rows. A bottom row counts from zero to a given n and the top counts from the right from n to 2n. The columns generated give all the whole numbers that add to 2n. We confirm that using a sieve, we do always seem to get top and bottom primes that show Goldbach's conjecture is true for the particular 2n depicted by this table. Next we cumulatively depict these tables and we see some interesting patterns. We can infer that all prime pairs will occur in one of these tables. We also see diagonal prime lines that seem to start and stop in symmetrical ways. These patterns suggest that for any even number we can choose a column and then find a prime pair.
[23]
vixra:2407.0169 [
pdf]
On Diophantine Equation Ax^4 + By^4 + Cz^4 + Dw^4 + Eu^4 = 0
In this paper, we prove that there are infinitely many integer solutions of ax^4 + by^4 + cz^4 + dw^4 + eu^4 = 0 where a+b+c+d+e=0.
[24]
vixra:2407.0143 [
pdf]
On Degrees of Carry and Scholz's Conjecture
Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2}(iota(n)-lfloor frac{log n}{log 2}floor+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}{frac{n}{2^j}})$$ then the inequality $$iota(2^n-1)leq n+1+sum limits_{j=1}^{lfloor frac{log n}{log 2}floor}bigg({frac{n}{2^j}}-xi(n,j)bigg)+iota(n)$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$, ${cdot}$ denotes the fractional part of $cdot$ and where $xi(n,1):={frac{n}{2}}$ with $xi(n,2)={frac{1}{2}lfloor frac{n}{2}floor}$ and so on
[25]
vixra:2407.0142 [
pdf]
[Attempted] Proof of Fermat's Conjecture in Just a Few Lines
We present in this paper a formula for decomposing a power of an integer into a product of consecutive integers and its properties. We also discuss properties of some specific vectors (polynomials). By using these concepts, we provide simple proofs for both of Fermat's theorems. Furthermore, the proof of the great Fermat theorem is accessible to all students who have studied the notion of vector space.
[26]
vixra:2407.0124 [
pdf]
On the L-Functions from Generalized Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, and the Prime Numbers from Polignac's and Twin Prime Conjectures
We analyze L-functions of elliptic curves (and apply Sign normalization) which support simplest version of Birch and Swinnerton-Dyer conjecture to be true. Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. The complete set with derived subsets of Odd Primes contain arbitrarily large number of elements and satisfy Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems being satisfied, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
[27]
vixra:2407.0092 [
pdf]
A Proof of the Collatz Conjecture
In this paper, we had given a proof of the Collatz conjecture in elementary algebra. Since any given positive integer is conjectured to return to odd 1 in operations, we analyze continuous inverse operations starting with odd 1, it had proved that all of the inverse path numbers of a given non-triple is obtainable and any inverse operation path tends to infinity, we can get any odd and even, to do continuous forward operations for a positive integer obtained it will return to the odd 1 along the inverse operation paths.
[28]
vixra:2407.0091 [
pdf]
[attempted] Proof for Goldbach’s Conjecture Verification with Mathematical Induction Formula
This document forwards a freshly unearthed test of the Goldbach Conjecture, a longstanding enigma in the theory of numbers put forth byChristian Goldbach in 1742. In our point of view, we have been able to come up with a simple and yet stunning explanation on how numberswhich are divisible by 2 could be permanently expressed as the sum of two prime numbers. Through an extensive analysis, it will be seen that every other two numbers above 2 can always be expressed in that manner. Our evidence is based on fundamental theories of numbers and original methods that solve the problem without any difficulty. Consequently, understandingis not difficult at all. The pathway for further research in number theory has just been brought to light while at the same time indicatinghow vital determination and a variation of outlook are for any endeavour.
[29]
vixra:2407.0055 [
pdf]
Prime Number Distribution Proving the Twin Prime and Goldbach Conjectures
The paper investigates the dispersion of prime numbers as well as the twin prime and goldbach’s conjectures. The initial key feature that primenumbers are never even (apart from 2) will be presented as the basis on which a new rule concerning their distribution can be developed. In that wise, this will help us to come up with a demonstration of why there exist an infinite number of odd pairs such that their difference is equal to 2. We also show that the Goldbach conjecture is true. This means that it is possible to write any even number greater than two as the sum of two prime numbers. The results contribute fresh knowledge concerning old mathematics subjects, especially those concerning the origins of prime numbers.
[30]
vixra:2407.0040 [
pdf]
Pi's Irrationality Using Maclaurin Polynomials
After reviewing Maclaurin series and the Alternating Series Estimation Theorem, we show how these can be combined with some arithmetic and algebraic observations to prove that pi is irrational.
[31]
vixra:2407.0019 [
pdf]
Transcendence of Deformations of Polylogarithm Functions in Non-Zero Characteristic
Let p be a prime number. In 2005, with the aim of reproving by Wade's method the transcendence for n ∈ Zp N of Γ∞(n), where Γ∞ is the Carlitz-Goss factorial, Yao introduces an uncountable class of deformations of Kochubei polylogarithm functions of which he shows the transcendence of their values u200bu200bin 1 (which is sufficient for his objective) and more generallyin 1/T^k (k ∈ N∗). In this present article, we answer Yao's question to extend this result into a non-zero algebraic. We prove this fact not only in the context of infinite place but also in that of finite places. We show an analogous result for the deformations of the polylogarithm function of Carlitz.
[32]
vixra:2406.0187 [
pdf]
Notes and Problems in Number Theory (Volume I)
This book is the first volume of a collection of notes and solved problems about number theory. Like my previous books, maximum clarity was one of the main objectives and criteria in determining the style of writing, presenting and structuring the book as well as selecting its contents.
[33]
vixra:2406.0154 [
pdf]
An Elementary Approach to X^2 +7 = 2^n
In this paper, we prove that the positive integer solutions of the equation x^2 +7 = 2^n are x = 1, 3, 5, 11, 181, corresponding to n = 3, 4, 5, 7, 15.
[34]
vixra:2406.0150 [
pdf]
Simple Symmetry Proves All Three Riemann's Hypotheses
Suppose the Riemann Zeta function is multiplied by two arbitrary functions, and the resulting functions' values are equated at symmetrical points concerning the critical line Re s = 1/2. In that case, the resulting system of fourequations has to give the positions of the Zeta function's zeros. However, since the functions are arbitrary, the positions of the zero places are arbitrary, making a zero coincide with non-zero. Hence, the Riemann Hypothesis that the only zeroes are those on the critical line is true. This simple text is proof of the Riemann hypothesis, Generalized Riemann hypothesis and Extended Riemann hypothesis with accordingfunctions.
[35]
vixra:2406.0070 [
pdf]
Analyzing Non-Trivial Zeros of the Riemann Zeta Function Using Polar Coordinates
This paper presents an approach to analyzing the non-trivial zeros of the Riemann zeta function using polar coordinates. We investigate whether the real part of all non-trivial zeros can be determined to be a constant value. By transforming the traditional complex plane into a polar coordinate system, we recalculated and examined several known non-trivial zeros of the zeta function. Our findings provide an alternative framework for understanding this profound mathematical conjecture.Through mathematical proof and leveraging analytic continuation and holomorphic function theory, we explore the nature of (sigma) in the polar coordinate system. This analysis transforms the problem into a geometric one, allowing for simpler and more intuitive calculations. This approach provides a step towards an alternative understanding of the properties of the Riemann zeta function's non-trivial zeros. The findings of this work indicates that wit this geometric perspective, the Riemann Hypothesis holds true.
[36]
vixra:2406.0045 [
pdf]
How to Solve Diophantine Equations
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in particular to young "mathematicians" dealing mostly with Diophantine equations at elementary level of number theory (noting that familiarity with elementary number theory is generally required).
[37]
vixra:2406.0020 [
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Discover a Proof of Goldbach’s Conjecture
This paper presents a new proof of the Goldbach conjecture, which is a well-known problem originating from number theory that was proposedby Christian Goldbach back in 1742. Our way gives a simple but deep understanding of the even integers can be written as the sum of two prime numbers Through examining fully we show that every other even integer larger than two will essentially represent itself in form adding up two prime numbers. The revelation of a straightforward and elegant line to this enduring conjecture comes from the use of basic number theory concepts such as; by going a step further and coming up with creative strategies. There is more evidence and sound payments she makes for her assertion as we continue.The centuries-old mathematical puzzle has been solved paving way for the exploration of new possibilities in number theory and we are grateful for the perspective and the persistence accorded us by God, which enabled us to reach this milestone.
[38]
vixra:2405.0159 [
pdf]
A New Approach to the Goldbach Conjecture
In 1742 Christian Goldbach suggested that any even number four or greater is the sum of two primes.The Goldbach conjecture remains unproven to the present day, although it has been verified for all even numbers up to 4 × 10^18. Previously this problem has been attacked using deep analytical methods and with complicated integer sieves. This paper takes an entirely new approach to the Goldbach conjecture using pairs of composite integers (composite pairs) that are used to find pairs of prime numbers (prime airs) that sum to the same even natural number.
[39]
vixra:2405.0129 [
pdf]
The abc Conjecture Is True
In this paper, we consider the abc conjecture. Assuming the conjecture c<rad^2(abc) is true, we give the proof of the abc conjecture for epsilon >1. For the case epsilon in ]0,1[, we consider that the abc conjecture is false, from the proof, we arrive in a contradiction.
[40]
vixra:2405.0048 [
pdf]
Both the 1-Dimensional Line from Sieve of Eratosthenes and 2-Dimensional Trajectory from Riemann Zeta Function Have Centroids as Perfect Point Symmetry
From perspective of Number theory, Dirichlet eta function (proxy function for Riemann zeta function as generating function for all nontrivial zeros) and Sieve of Eratosthenes (as generating algorithm for all prime numbers) are essentially infinite series. We apply infinitesimals to their outputs. Riemann hypothesis asserts the complete set of all nontrivial zeros from Riemann zeta function is located on its critical line. It is proven to be true when usefully regarded as an Incompletely Predictable Problem. We ignore even prime number 2. The complete set with derived subsets of Odd Primes all contain arbitrarily large number of elements while satisfying Prime number theorem for Arithmetic Progressions, Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. Having these theorems satisfied by all Odd Primes, Polignac's and Twin prime conjectures are separately proven to be true when usefully regarded as Incompletely Predictable Problems.
[41]
vixra:2405.0035 [
pdf]
The Riemann Hypothesis Has Three Types of Non Trivial Zeros
This paper classifies non trivial zeros based on the Riemann Zeta function. Through this operation, we can clearly understand the distribution pattern of non trivial zeros and predict the position of the next zero point. You can know that the Riemann hypothesis has three types of non trivial zeros, and the first type of non trivial zeros is located on the critical line, while the second and third types of zeros are not. Meanwhile, through a series of equation derivations, we can also understand why it is so difficult to find counterexamples of the Riemann hypothesis.
[42]
vixra:2405.0024 [
pdf]
A Proof of Twin Prime Conjecture by Clement’s Theorem
I proved the Twin Prime Conjecture by using Clement’s theorem. I was able to transform below. (n is positive integer) 4×(6n − 2)! + 6n + 3 ≡ 0 (mod (6n − 1)(6n + 1)). Even if the number(n) reaches the limit, use n=x+1,n=x+2...n=x+18 from n=x. By n=x+18, new twin prime numbers are found.In this way, even larger twin primes are born.Repeat this. That is, Twin Primes exist forever.
[43]
vixra:2404.0054 [
pdf]
The Density of Minimal Dividing Odd Subsets for the Even Numbers is Asymptotically Normal
In this notice, we introduce the problem of minimal dividing odd subsets for the even numbers and we show that the density of such subsets of $n$ elements is asymptotically normal (that is at least decreasing as $frac{1}{n}$). We argue that understanding the problem of minimal dividing oddsubset might lead to new approaches to solving NP-hard problems.
[44]
vixra:2404.0040 [
pdf]
Proofs for Collatz Conjecture and Kaakuma Sequence
The objective of this study is to present precise proofs of the Collatz conjecture and introduce some interesting behavior on Kaakuma sequence. We propose a novel approach that tackles the Collatz conjecture using different techniques and angles. The Collatz Conjecture, proposed by Lothar Collatz in 1937, remains one of the most intriguing unsolved problems in mathematics. The conjecture posits that, for any positive integer, applying a series of operations will eventually lead to the number 1. Despite decades of rigorous investigation and countless computational verification, a complete proof has eluded mathematicians. In this research endeavor, we embark on a comprehensive exploration of the Collatz Conjecture, aiming to shed light on its underlying principles and ultimately establish its validity. Our investigation begins by defining the Collatz function and investigating some behavior like. transformation, selective mapping, successive division, constant growth rate of inverse tree, and more. Using and analyzing the discovered properties of Collatz sequence we can show there are contradiction. In addition to this we investigate Qodaa ratio test that validates the reality of discovered behavior of Collatz sequence and works for infinite and distinct Kaakuma sequences. Our investigation culminates in the formulation of a set of conjectures encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz Conjecture and contribute to our understanding of the underlying mathematical structure. This proof helps to change some researchers' views on unsolved problems and offers new perspectives on probability in infinite range. In this study, we uncover the dynamic nature of the Collatz sequence and provide a reflection and interpretation of the probabilistic proof of the Collatz Conjecture.
[45]
vixra:2404.0037 [
pdf]
Contribution to Lonely Runner Conjecture
The Lonely Runner conjecture finds its mathematical description in winding the runner's linear paths into the complete cycle, c-cycle, on the unit track circle. All runners, to finish the competition, must complete the c-cyclesimultaneously. Any collection BT mR_{cn}ET of the BT cnET integer speeds runners and maximum speed BT v_{cn} =nET is a subset of the enveloping collection BT mR_{n}={ 1,2,3,cdots,n}ET of the BT n,;n>cn,ETrunners with the maximum speed BT n.ET[0.50mm]The time period BT 1_{ct}=1/nET of the fastest runner, f-runner BT n,ET defines the set of BT nET right half open time f-segments of the measure BT 1_{ct},ET which cover the c-cycle time domain of the measure BT 1.ETThe winding mapping of the linear paths BT X(t)ET associates with the c-cycle Graph BT G,ET the union of the BT nET individual graphs BT g=(t,X(t)),ET reduced to the domain of the c-cycle. The time domain segmentation partitions the Graph BT GET into BT nET Subgraphs BT G_{i},ET each one on the one of the BT nET f-segments. The final Subgraph bundle sinks into the point BT (1,1).ETAt the end of the first f-segment, all the runners arranged into f-constellation at the BT nET fixed, stationary points on the unit circle in the sequence of the increasing speeds. However, at the final f-segment, the runners, on the way to the starting point, are arranged at the decreasing speed order at the same stationary points.The speed order inversion inverts the slope order of the graphs on the final Subgraph bundle.Finally, the infimum graph BT g_{n-1}ET of the Subgraph bundle of the BT n-1ET runner's mutual separation graphs, the graph of thelargest slope connects the points BT (0,n-1)1_{cl}ET and BT (1,1)1_{cl}.ETConsequently, the Lonely Runner conjecture is true on the set BT mR_{n},ET and must be true on any of its subset BT mR_{cn},ET
[46]
vixra:2404.0016 [
pdf]
The Extension of the Riemann's Zeta Function
Prime numbers [See 1-7] are used especially in information technology, such as public-key cryptography , and recall that the distribution of prime numbers is closely related to the non-trivial zeros of the zeta function therefore related to the Riemann hypothesis.Here I introduce the function $circledS$: $ (X,z) longmapsto prod_{pinmathcal{P}} frac1{1-X/p^{z}} $ which is a generalization of the function $zeta $ of Riemann that I will use to prove the Riemann hypothesis.
[47]
vixra:2403.0097 [
pdf]
Merging the Goldbach and the Bunyakovsky Conjecture into a Unified Prime Axiom of Second-Order Logic and Investigating Much Beyond the Goldbach's Conjecture and the Prime Number Theorem
Merging the Goldbach and the Bunyakovsky conjecture into a Unified Prime Axiom of second-order logic and investigating much beyond the Goldbach's conjecture and the prime number theorem.
[48]
vixra:2403.0077 [
pdf]
The Collatz Conjecture, Pythagorean Triples, and the Riemann Hypothesis: Unveiling a Novel Connection Through Dropping Times
In the landscape of mathematical inquiry, where the ancient and the modern intertwine, few problems captivate the imagination as profoundly as the Collatz conjecture and the quest for Pythagorean triples. The former, a puzzle that has defied solution since its inception in the 1930s by Lothar Collatz, asks us to consider a simple iterative process: for any positive integer, if it is even, divide it by two; if it is odd, triple it and add one. Despite its apparent simplicity, the conjecture leads us into a labyrinth of diverse complexity, where patterns emerge and dissolve in an unpredictable dance. On the other hand, Pythagorean triples, sets of three integers that satisfy the ancient Pythagorean theorem, have been a cornerstone of geometry since the time of the ancient Greeks, embodying the harmony of numbers and the elegance of spatial relationships. This exploratory paper embarks on an unprecedented journey to bridge these seemingly disparatedomains of mathematics. At the heart of this exploration is the discovery of a novel connection between Collatz dropping times and Pythagorean triples. I will demonstrate how the dropping time of each odd number can be uniquely associated with a Pythagorean triple. As you will see, the triples seem to be encoding spatial information about Collatz trajectories. As we begin to work with triples, we’ll be motivated to move from the number line to the complex plane where we find structure andbehavior resembling that of the Riemann Zeta function and it’s zeros.
[49]
vixra:2402.0163 [
pdf]
Isolating the Prime Numbers
Prime numbers greater than 3 belong to the number sequences 6n ± 1 wheren ≥ 1. These sequences also include the composites that are not divisible by 2 and/or3 and therefore their factors must also be of the form 6n ± 1. This allows all of the6n±1 composites to be equivalently written in the form of factors (6n1±1)(6n2±1),where n1 and n2 ≥ 1, creating three sub-sequences that exclude prime numbers.Finding and isolating the prime numbers can be achieved by selecting a numberrange and creating a set of 6n ± 1 numbers for that range before subtracting thesubsets (6n1 ± 1)(6n2 ± 1) to isolate and identify all the primes in the set.
[50]
vixra:2402.0115 [
pdf]
The First Counterexample of Riemann Hypothesis Found Through Computer Calculation
The counterexample of the Riemann hypothesis causes a significant change in the image of the Riemann Zeta function, which can be distinguished using mathematical judgment equations. The first counterexample can be found through this equation.
[51]
vixra:2402.0110 [
pdf]
Further Investigations on Euler's Odd Perfect Numbers
It is a long-standing question whether there exists an odd perfect number. This article establishes a complete theory in order to prove that if an oddperfect number n exists then n = pm^2 with p prime and p is congruent to 1 (mod 4), andgcd (p, m) = 1.
[52]
vixra:2402.0091 [
pdf]
The Signature of Abc Conjecture is Proven
Equivalent view of abc conjecture is proven. Some crucial properties of the abc conjecture are presented and proven. For example, there exist three numbers (a, b, c) that satisfy the abc conjecture for an arbitrary value c.
[53]
vixra:2402.0081 [
pdf]
A New Attempt to Check Whether Ramanujan's Formula Pi^4 Approx 97.5-1/11 is a Part of Some Completely Accurate Formula
Intuitively, it seems that Ramanujan's formula $pi^4approx 97.5-1/11$ is an approximation for some perfectly accurate formula for $pi$. Here is one attempt to prove this. The principle of proof, however, is based on closeness of the every rest term to the inverse of integers. Although it is indeed somewhat closer to integers than it is on average, this proof is not complete. So we cannot say for sure whether this proves or disproves that this Ramanujan's formula has higher approximations; however, it gives hints and opens up space for further research.Moreover, this attempted proof is quite original. Also, such a method could also help in physics.
[54]
vixra:2402.0058 [
pdf]
Validating Collatz Conjecture Through Binary Representation and Probabilistic Path Analysis
The Collatz conjecture, a longstanding mathematical puzzle, posits that, regardless of the starting integer, iteratively applying a specific formula will eventually lead to the value 1. This paper introduces a novelapproach to validate the Collatz conjecture by leveraging the binary representation of generated numbers. Each transition in the sequence is predetermined using the Collatz conjecture formula, yet the path of transitionsis revealed to be intricate, involving alternating increases and decreases for each initial value. The study delves into the global flow of the sequence, investigating thebehavior of the generated numbers as they progress toward the termination value of 1. The analysis utilizes the concept of probability to shed light on the complex dynamics of the Collatz conjecture. By incorporatingprobabilistic methods, this research aims to unravel the underlying patterns and tendencies that govern the convergence of the sequence.The findings contribute to a deeper understanding of the Collatz conjecture,offering insights into the inherent complexities of its trajectories. This work not only validates the conjecture through binary representation but also provides a probabilistic framework to elucidate the global flow ofthe sequence, enriching our comprehension of this enduring mathematical mystery.
[55]
vixra:2402.0024 [
pdf]
Adaptive Polynomial Factorization (APF) Method: Enhanced Factorization using Modified Pollard Rho Algorithm
This research paper introduces the Adaptive Polynomial Factorization (APF) Method, an enhanced factorization technique based on the Modified Pollard Rho Algorithm. The method incorporates adaptive polynomial evaluation, providing efficiency in factorization tasks. The paper presents a mathematical representation, performance analysis, and examples showcasing the APF Method’s versatility and superiority over the original Pollard Rho algorithm.
[56]
vixra:2402.0020 [
pdf]
Beyond Fibonacci: the Sequence of Integer Powers of Numbers
Neighboring triangles of Pascal's triangle, cousin numbers of the golden ratio, a simplified formula giving the numbers of generalized Fibonacci sequences, associated generating function, chaos theory and tent function equation.<p>Triangles voisins du triangle de Pascal, nombres cousins du nombre d'or, une formule simplifiée donnant les nombres des suites de Fibonacci généralisées, fonction génératrice associée, théorie du chaos et équation de la fonction tente.
[57]
vixra:2402.0003 [
pdf]
Analyzing the Connection from (H(z)) to the Riemann Zeta Function
This paper explores the intriguing connection between the function (H(z) = ln(|sec(pi z/log(z))|)) and the Riemann Zeta Function (zeta(s)). The journey begins by investigating the zeros of (H(z)) and employing advanced mathematical tools such as the Taylor series expansion, the argument principle, and the inverse Mellin transform. Through this exploration, we establish a relationship that leads to a complex integral representation connecting (H(z)) to the Riemann Zeta Function (zeta(s)).
[58]
vixra:2401.0132 [
pdf]
Periodic Function Approach to Prime Number Analysis with Graphical Illustrations
This paper introduces a novel approach employing periodic functions for the comprehensive analysis of prime numbers. The method ncompassesprimality testing, factor counting and listing, prime distribution calculation, and the determination of the Nth prime. The exposition of the technique is presented in a clear and sequential manner, guiding the reader through each step with explicit equations. Graphs are strategically incorporated between crucial stages to facilitate a rapid and intuitive visualization of the rationale and outcomes of each maneuver. The paper concludes with concise reflections and ongoing inquiries into the potential applications and refinements of the proposed method.
[59]
vixra:2401.0081 [
pdf]
Hypothesis: Distribution of Primes and the Logarithmic Expression
This research explores the distribution of prime numbers using a novel ljogarithmic expression. The hypothesis suggests that an expression, partitions natural numbers into groups, revealing a systematic distribution of primes. Experimental results demonstrate an intriguing pattern as the range of N increases, with the average number of primes in each group stabilizing around 15. The paper discusses thebackground, mathematical expression, experimental results, and potentialavenues for future research.
[60]
vixra:2401.0068 [
pdf]
A Convergent Subsequence of $theta_n(x+iy)$ in a Half Strip
For $frac{1}{2}<x<1$, $y>0$ and $ninmathbb{N}$, let $displaystyletheta_n(x+iy)=sum_{i=1}^nfrac{{mbox{sgn}}, q_i}{q_i^{x+iy}}$,where $Q={q_1,q_2,q_3,cdots}$ is the set of finite product of distinct odd primes and${mbox{sgn}}, q=(-1)^k$ if $q$ is the product of $k$ distinct primes.In this paper we prove that there exists an ordering on $Q$ such that $theta_n(x+iy)$ has a convergent subsequence.
[61]
vixra:2401.0054 [
pdf]
On the Sum of Reciprocals of Primes
Suppose that $y>0$, $0leqalpha<2pi$ and $0<K<1$. Let $P^+$ be the set of primes $p$ such that $cos(yln p+alpha)>K$ and $P^-$ the set of primes $p$ such that $cos(yln p+alpha)<-K$ . In this paper we prove $sum_{pin P^+}frac{1}{p}=infty$ and $sum_{pin P^-}frac{1}{p}=infty$.
[62]
vixra:2401.0046 [
pdf]
Redefining Mathematical Structure: From the Real Number Non-Field to the Energy Number Field
The traditional classification of real numbers (R) as a complete ordered field is contested throughcritical examination of the field axioms, with a focus on the absence of a multiplicative inverse for zero. We propose an alternative mathematical structure based on Energy Numbers (E), deriving from quantum mechanics, which addresses the classical anomalies and fulfills field properties universally, including an element structurally analogous but functionally distinctive from the zero in R.
[63]
vixra:2401.0009 [
pdf]
Contribution to Goldbach's Conjectures
The internal structure of the natural numbers reveals the relation between the weak and strong Goldbach's conjectures. Explicitly, if the weak Goldbach's conjecture is true, the strong Goldbach's conjecture is, and Goldbach's conjectures are true.
[64]
vixra:2401.0008 [
pdf]
Goldbach's Number Construction
The internal structure of the natural numbers reveals the relation between the weak and the strong Goldbach's conjectures. The three prime integers structure of the odd integers alreadycontains the two prime integers base of the even integers. An explicit one-to-one correspondence between these two structures, defined asGoldbach's numbers exist. Thus, if the weak Goldbach's conjecture is true, the strong Goldbach'sconjecture should be. Hopefully, this will bring a happy end to Goldbach'sconjecture problem.
[65]
vixra:2312.0157 [
pdf]
Investigation on Brocard-Ramanujan Problem
Exploring n! + 1 = m^2 for natural number solutions beyond n = 4, 5, 7 confirms no further solutions exist,validated by using GCD Linear Combination Theorem
[66]
vixra:2312.0143 [
pdf]
On the Nonexistence of Solutions to a Diophantine Equation Involving Prime Powers
This paper investigates the Diophantine equation pr + (p + 1)s = z2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.
[67]
vixra:2312.0135 [
pdf]
On the Notion of Carries of Numbers 2^n-1 and Scholz Conjecture
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$kappa(2^n-1)=frac{1}{2(1+c)}lfloor frac{log n}{log 2}floor-1$$ for $c>0$ fixed, then the inequality $$iota(2^n-1)leq n-1+(1+frac{1}{1+c})lfloorfrac{log n}{log 2}floor$$ holds for all $nin mathbb{N}$ with $ngeq 4$, where $iota(cdot)$ denotes the length of the shortest addition chain producing $cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$kappa(2^n-1):=(frac{1}{1+f(n)})lfloor frac{log n}{log 2}floor-1$$ with $f(n)=o(log n)$ and $f(n)longrightarrow infty$ as $nlongrightarrow infty$ for $ngeq 4$ then the inequality $$iota(2^n-1)leq n-1+(1+frac{2}{1+f(n)})lfloorfrac{log n}{log 2}floor$$ holds.
[68]
vixra:2312.0134 [
pdf]
A Proof of the Wen-Yao Conjecture
In this article, we characterize monomials in de facto values.Carlitz-Goss rielle defined on the complement of Fq (T) in a finite place which arealgebraic on Fq (T ). In particular, this confirms Wen-Yao's conjecturestated in 2003. This gives a necessary and sufficient condition on an en-p-adic tier so that the value of the Carlitz-Goss factorial in it is algebraic on Fq (T ). When restricted to rational arguments, we determinenot all algebraic relations between the values u200bu200btaken by this function, this which gives the counterpart for finite places of a result of Chang, Papanikolas, Thakur and Yu obtained in the case of infinite place.<p>Dans cette article, nous caractérisons les monômes en les valeurs de la facto-rielle de Carlitz-Goss définie sur le complété de Fq (T ) en une place finie qui sont algébriques sur Fq (T ). En particulier, cela confirme la conjecture de Wen-Yaoénoncée en 2003 . Celle-ci donne une condition necessaire et suffisante sur un en-tier p-adique pour que la valeur de la factorielle de Carlitz-Goss en celui-ci soit algébrique sur Fq (T ). Lorsque restreint aux arguments rationnels, nous détermi-nons toutes les relations algébriques entre les valeurs prises par cette fonction, cequi donne le pendant pour les places finies d’un résultat de Chang, Papanikolas, Thakur et Yu obtenu dans le cas de la place infinie.
[69]
vixra:2312.0108 [
pdf]
Complete Operations
The Operator axioms have produced complete operations with real operators. Numerical computations have been constructed for complete operations. The classic calculator could only execute 7 operator operations: + operator operation(addition), - operator operation(subtraction), $times$ operator operation(multiplication), $div$ operator operation(division), ^{} operator operation(exponentiation), $surd$ operator operation(root extraction), log operator operation(logarithm). In this paper, we invent a complete calculator as a software calculator to execute complete operations. The experiments on the complete calculator could directly prove such a corollary: Operator axioms are consistent.
[70]
vixra:2312.0036 [
pdf]
New Equivalent of the Riemann Hypothesis
In this article, it is demonstrated that if the zeta function does not have a sequence of zeros whose real part converges to 1, then it cannot have any zeros in the critical strip, showing that the Riemann Hypothesis is false.
[71]
vixra:2311.0137 [
pdf]
New Bounds on Mertens Function
In this brief paper we study and bound Mertens function. The main breakthrough is the obtention of a Möbius-invertible formulation of Mertens function, which with some transformations and the application of a generalization of Möbius inversion formula, allows us to reach an asymptotic rate of growth of Mertens function that proves the Riemann Hypothesis.
[72]
vixra:2311.0119 [
pdf]
Zeta Function
This article delves into the properties of the Riemann zeta function, providing a demonstration of the existence of a sequence of zeros ${z_k}$, where $lim operatorname{Re}(z_k) = 1$. The exploration of these mathematical phenomena contributes to our understanding of complex analysis and the behavior of the zeta function on the critical line.
[73]
vixra:2311.0086 [
pdf]
On the Largest Prime Factor of the K-Generalized Lucas Numbers
Let $(L_n^{(k)})_{ngeq 2-k}$ be the sequence of $k$--generalized Lucas numbers for some fixed integer $kge 2$ whose first $k$ terms are $0,ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. For an integer $m$, let $P(m)$ denote the largest prime factor of $m$, with $P(0)=P(pm 1)=1$. We show that if $n ge k + 1$, then $P (L_n^{(k)} ) > (1/86) log log n$. Furthermore, we determine all the $k$--generalized Lucas numbers $L_n^{(k)}$ whose largest prime factor is at most $ 7$.
[74]
vixra:2311.0059 [
pdf]
Divisible Cyclic Numbers
There are known to exist a number of (multiplicative) cyclic numbers, but in this paper I introduce what appears to be a new kind of number, which we call divisible cyclic numbers (DCNs), examine some of their properties and give a proof of their cyclic property. It seems remarkable that I can find no reference to them anywhere. Given their simplicity, it would be extraordinary if they were hitherto unknown.
[75]
vixra:2311.0052 [
pdf]
On the Incompletely Predictable Problems of Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures
We validly ignore even prime number 2. Based on all arbitrarily large number of even prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with Prime number theorem for Arithmetic Progressions. With this condition being satisfied by all Odd Primes, we argue that Modified Polignac's and Twin prime conjectures are proven to be true when these conjectures are treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (which acts as proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when this hypothesis is treated as Incompletely Predictable Problem.
[76]
vixra:2311.0050 [
pdf]
Mathematics for Incompletely Predictable Problems Required to Prove Riemann Hypothesis, Modified Polignac's and Twin Prime Conjectures
As two different but related infinite-length equations through analytic continuation, Hasse principle is satisfied by Riemann zeta function as a certain type of equation that generates all infinitely-many trivial zeros but this principle is not satisfied by its proxy Dirichlet eta function as a dissimilar type of equation that generates all infinitely-many nontrivial zeros. Based on two seemingly different location that are in fact identical, all nontrivial zeros are mathematically located on critical line or geometrically located on Origin point. Thus we prove location for complete Set nontrivial zeros to be critical line confirming Riemann hypothesis to be true. Sieve of Eratosthenes as a certain type of infinite-length algorithm is exactly constituted by an Arbitrarily Large Number of (self-)similar infinite-length sub-algorithms that are specified by every even Prime gaps. Modified Hasse principle is satisfied by this algorithm and its sub-algorithms that perpetually generate the Arbitrarily Large Number of all Odd Primes. Thus we prove Set even Prime gaps with corresponding Subsets Odd Primes all have cardinality Arbitrarily Large in Number confirming Modified Polignac's and Twin prime conjectures to be true.
[77]
vixra:2311.0030 [
pdf]
Euler's Identity, Leibniz Tables, and the Irrationality of Pi
Using techniques that show show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's formula. The proof involves evaluation of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.
[78]
vixra:2311.0015 [
pdf]
Reverse Chebyshev Bias in the Distribution of Superprimes
We study the distribution of superprimes, a subsequence of prime numbers with prime indices, mod 4. Rather unexpectedly, this subsequenceexhibits a reverse Chebyshev bias: terms of the form 4k + 1 are more common than those of the form 4k + 3, whereas the opposite is the case in the sequence of all primes. The effect, while initially weak and easy to overlook, tends tobe several times larger than the Chebyshev bias for all primes for samples of comparable size, at least, by one simple measure. By two other measures, it can be seen as fairly strong; by the same measures the ordinary Chebyshev effectis very strong. Both of these measures also imply that the reverse Chebyshev bias for superprimes is more volatile than the ordinary Chebyshev bias.
[79]
vixra:2310.0133 [
pdf]
Guessing that the Riemann Hypothesis is Unprovable Using Non-Standard Analysis
Riemann Hypothesis has been the unsolved conjecture for 164 years. This conjecture is the last one of conjectures without proof in "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse"(B.Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2. Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper, I guess the independence (unprovability) of the Riemann Hypothesis.
[80]
vixra:2310.0113 [
pdf]
New Maximum Interval Between Any Number and the Nearest Prime Number and Related Conjectures
In this short paper we prove that for n ≥ 2953652287 it exists some prime number between nand n + log(n), improving the best known proved bounds for the maximum interval between anynumber and the nearest prime number, as well as the maximum difference between two consecutiveprime numbers (prime gap). We note that this result proves some open conjectures on prime gapsand maximum intervals between any number and the nearest prime number.
[81]
vixra:2310.0046 [
pdf]
The Philosophical and Mathematical Implications of Division by 0/0 = 1 in Light of Einstein’s Theory of Special Relativity
The enigma of dividing zero by zero 0 0 has perplexed scholars across philosophy, mathematics, and physics, remaining devoid of a clear-cut solution. This lingering conundrum leaves us in an unsatisfactory position,as there emerges a genuine necessity for such divisions, particularly in scenarios involving tensor components that are both set at zero. This article endeavors to grapple with this profound issue by leveraging the insights of Einstein’s theory of special relativity. Surprisingly, when we wholeheartedly embrace the ramifications of this theory, it becomes evidentthat zero divided by zero must equate to one 00 = 1. Essentially, we are confronted with a pivotal decision: either embrace the feasibility and definition of dividing zero by zero, in accordance with Einstein’s theory of special relativity, or reevaluate the integrity of this fundamental theory itself. This exploration delves into the profound consequences arisingfrom this critical choice.
[82]
vixra:2310.0041 [
pdf]
Exist[ence Of] a Prime in Interval N^2 and "N^2+epsilon n"
Oppermance’ conjecture states that there is a prime number between n^2 and n^2 + n for every positive integer n,first we show that , All integer numbers between x^2 and x^2 + ϵx can be written as x^2 + i > 4p that 1 ≤ i ≤ ϵx andp = (x − m − 2)2 + j in which j is a number in intervals 1 ≤ j ≤ ϵ(x − m − 2),and then we prove generalization of Oppermance’ conjecture i.e there is a prime number in interval n^2 and n^2 + ϵn such that 0 < ϵ ≤ 1.
[83]
vixra:2309.0109 [
pdf]
The Geometric Collatz Correspondence
The Collatz Conjecture, one of the most renowned unsolved problems in mathematics, presents adeceptive simplicity that has perplexed both experts and novices. Distinctive in nature, it leaves manyunsure of how to approach its analysis. My exploration into this enigma has unveiled two compellingconnections: firstly, a link between Collatz orbits and Pythagorean Triples; secondly, a tie to theproblem of tiling a 2D plane. This latter association suggests a potential relationship with PenroseTilings, which are notable for their non-repetitive plane tiling. This quality, reminiscent of theunpredictable yet non-repeating trajectories of Collatz sequences, provides a novel avenue to probethe conjecture’s complexities. To clarify these connections, I introduce a framework that interpretsthe Collatz Function as a process that maps each integer to a unique point on the complex plane.In a curious twist, my exploration into the 3D geometric interpretation of the Collatz Function has nudged open a small, yet intriguing door to a potential parallel in the world of physics. A subtle link appears to manifest between the properties of certain objects in this space and the atomic energy spectral series of hydrogen, a fundamental aspect in quantum mechanics. While this connection is in its early stages and the depth of its significance is yet to be fully unveiled, it subtly implies a simple merging where pure mathematics and applied physics might come together.The findings in this paper have led me to pursue development of a new type of number I call a Cam number, which stands for "complex and massive", indicating that it is a number with properties that on one hand act like a scalar, but on the other hand act as a complex number. Cam numbers can be thought of as having somewhat dual identities which reveal their properties and behavior under iterations of the Collatz Function. This paper serves as a motivator for a pursuit of a theory of Cam numbers.
[84]
vixra:2309.0088 [
pdf]
Collatz Conjecture Proof for Special Integer Subsets and a Unified Criterion for Twin Prime Identification
This paper presents a proof of the Collatz conjecture for a specific subset of positive integers, those formed by multiplying a prime number "p" greater than three with an odd integer "u" derived using Fermat’s little theorem. Additionally, we introduce a novel screening criterion for identifying candidate twin primes, extending our previous work linking twin primes (p and p+2) with the equation 2(p−2) = pu+v, where unique solutions for u and v are required. This unified criterion offers a promising approach to twin prime identification within a wider range of integers, further advancing research in this mathematical domain.
[85]
vixra:2309.0062 [
pdf]
Expressing Even Numbers Beyond 6 as Sums of Two Primes
The "strong Goldbach conjecture" posits that any even number exceeding 6 can be represented as the sum of two prime numbers. This study explores this hypothesis, leveraging the constancy of odd integer quantities and cumulative sums within positive integers. By identifying odd prime numbers, pα1and pα2, within [3, n] and (n, 2n-2) intervals, we demonstrate a transformative process grounded in the unchanging nature of odd number counts and their cumulative sums. Through this process, we establish the equation 2n =pα1 + pα2, offering a significant stride in unraveling the enigmatic core of the strong Goldbach conjecture.
[86]
vixra:2309.0049 [
pdf]
The Sum of Positive and Negative Prime Numbers are Equal
This paper unveils a profound equation that harnesses the power of natural numbers to establish a captivating theorem: the balance between positive and negative prime numbers’ summation, intricately linked through the medium of natural numbers. As a corollary, the essence ofnatural numbers emerges as a testament to the harmonious interplay between even and odd elements. Notably, we expose the remarkable revelation that odd numbers find expression as both the aggregate of prime divisors and the sum of prime numbers, fusing diverse mathematical concepts into an elegant unity. This work reshapes the landscape of number theory, illuminating the hidden connections between primes, naturals, andtheir arithmetical compositions.
[87]
vixra:2309.0043 [
pdf]
On Bifurcations and Beauty
This paper focuses on two ideas: the beginning focuses on standard and chaotic bifurcations, and the end focuses on beauty through mathematical coincidences. The scope of the bifurcation side is ambitious: relating bifurcation theory not only to the logistic map but also to prime spirals, the Riemann hypothesis, the Lambert W function, the Collatz conjecture, the Mandelbrot set, and music theory. The scope of the beauty side is similar: a proposed sequence that is opposite to the primes in some sense, finite sequences with peculiar properties, Fibonacci-like sequences, trees of primitive Pythagorean triples, Babylonian math, Grimm's conjecture, and Shell sort. Rather than providing rigorous analysis, my goal is to revitalize qualitative mathematics.
[88]
vixra:2309.0020 [
pdf]
Hilbert and Pólya Conjecture, Dynamical System, Prime Numbers, Black Holes, Quantum Mechanics, and the Riemann Hypothesis
In mathematics, the search for exact formulas giving all the prime numbers, certain families of prime numbers or the n-th prime number has generally proved to be vain, which has led to contenting oneself with approximate formulas [8]. The purpose of this article is to give a simple function to produce the list of all prime numbers.And then I give a generalization of this result and we show a link with the quantum mechanics and the attraction of black Holes. And I give a new proof of lemma 1 which gave a proof of the Riemann hypothesis [4]. Finally another excellent new proof o f the Riemann hypothesisis given and I deduce the proof of Hilbert Polya's conjecture
[89]
vixra:2308.0197 [
pdf]
On the Existence of Solutions to Erdh{o}s-Straus Type Equations
We apply the notion of the textbf{olloid} to show that the family of ErdH{o}s-Straus type equation $$frac{4^{2^l}}{n^{2^l}}=frac{1}{x^{2^l}}+frac{1}{y^{2^l}}+frac{1}{z^{2^l}}$$ has solutions for all $lgeq 1$ provided the equation $$frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solution for a fixed $n>4$.
[90]
vixra:2308.0177 [
pdf]
Natural Number Infinite Formula and the Nexus of Fundamental Scientific Issues
Within this paper, we embark on a comprehensive exploration of the profound scientific issues intertwined with the concept of the infinitewithin the realm of natural numbers. Through meticulous analysis, we delve into three distinct perspectives that shed light on the nature of natural number infinity. By considering the framework of time reference, we confront and address the inherent challenges that arise when contemplating the infinite. Furthermore, we navigate the intricate relationship betweenthe infinite and fundamental scientific questions, seeking to unveil novel insights and resolutions. In a departure from conventional viewpoints,our examination of natural number infinity takes on a relativistic dimension, scrutinizing the role of time and the observer’s perspective. Strikingly, as we delve deeper into the foundational strata, we uncover the pivotal significance of relativity not only in physics but also in mathematics. This realization propels us towards a more holistic and consistentmathematical framework, underlining the inextricable link between the infinitude of natural numbers and the essential constructs of time and perspective.
[91]
vixra:2308.0164 [
pdf]
The Simple Structure of Prime Numbers
The prime numbers have a pseudo-random structure. And this structure is not simple. In this paper, we analyze the behavior of prime numbers. And we diagnose the inner body of the prime numbers.
[92]
vixra:2308.0131 [
pdf]
Exact Sum of Prime Numbers in Matrix Form
This paper introduces a novel approach to represent the nth sum of prime numbers using column matrices and diagonal matrices. The proposed method provides a concise and efficient matrix form for computing and visualizing these sums, promising potential insights in number theory and matrix algebra. The innovative representation offers a new perspectiveto explore the properties of prime numbers in the context of matrix algebra.
[93]
vixra:2308.0130 [
pdf]
Connected Old and New Prime Number Theory with Upper and Lower Bounds
In this article, we establish a connection between classical and modern prime number theory using upper and lower bounds. Additionally, weintroduce a new technique to calculate the sum of prime numbers.
[94]
vixra:2308.0063 [
pdf]
A Conjecture On σ(n) Function
We know many Arithmetical Functions [1] like ϕ(n),σ(n),τ(n) etc. In this paper we will discuss about σ(n) and will see a phenomenal observation.And later we will claim this observation as a conjecture.
[95]
vixra:2308.0056 [
pdf]
Unexpected Connection Between Triangular Numbers and the Golden Ratio
We find out that when a sum of five consecutive triangular numbers, $S_5(n)= T(n)+...+T(n+4)$, is also a triangular number $T(k)$, the ratios of consecutive terms of $a(i)$ that represent values of $n$ for which this happens, tend to $phi^2$ or $phi^4$ as $i$ tends to infinity, where $phi$ is the Golden Ratio. At the same time, the ratios of consecutive terms $S_5(a(i))$ tend to $phi^4$ or $phi^8$. We also note that such ratios that are the powers of $phi$ can appear in the sequences of triangular numbers that are also higher polygonal numbers, one case of which are the heptagonal triangular numbers.
[96]
vixra:2308.0025 [
pdf]
Analytic Proof of The Prime Number Theorem
In this paper, we shall prove the textit{Prime Number Theorem} by providing a brief introduction about the famous textit{Riemann Zeta Function} and using its properties.
[97]
vixra:2308.0024 [
pdf]
Quasi-Perfect Numbers Have at Least 8 Prime Divisors
Quasi-perfect numbers satisfy the equation sigma(N) = 2*N+1, where sigma is the divisor summatory function. By computation, it is shown that no quasi-perfect number has less than 8 prime divisors. For testing purposes, quasi-multiperfect numbers are examined also.
[98]
vixra:2307.0129 [
pdf]
A Proof of the Legendre Conjecture
If Legendre conjecture does not hold all integers in the interior of BT (n^{2},(n+1)^{2})ET are composed numbers. The composite integers counting shown that the rate of the number of the odd composites to the number of odd integers in theinterior of BT (n^{2},(n+1)^{2})ET is smaller than one. Consequently, the Legendre conjecture holds.
[99]
vixra:2307.0086 [
pdf]
Riemann Hypothesis Proof
The Riemann Hypothesis, a famous unsolved problem in mathematics, posits a deep connection between the distribution of prime numbers and the nontrivial zeros of the Riemann zeta function. In this study, we investigate the presence of zeros at prime numbers in a specific mathematical expression, $ln (sec (pi cdot nlog(n)))$, and its implications for the Riemann hypothesis. By employing rigorous mathematical analysis, we establish a clear connection between prime numbers, trigonometric functions, and the behavior of the Riemann zeta function. Our findings contribute to the body of knowledge surrounding the Riemann hypothesis and its potential proof, shedding light on the intricate nature of prime numbers and their relationship to fundamental mathematical functions.
[100]
vixra:2306.0135 [
pdf]
The Elemental Property of Primes and Small Gaps Between Primes
The solution to the Twin Prime Conjecture lies in the elemental property of primes. We construct a sequence of consecutive primes, analyzing and handling them by the combination of the elemental property of primes and the Statistics theory reveal that Twin Prime Conjecture is true.
[101]
vixra:2306.0061 [
pdf]
İkiz Asallar Kestirimi İspatı (Proof for Twin Prime Conjecture)
İkiz asallar, aralarındaki fark 2 olan asal sayılardır. Sonsuz sayıda ikiz asal sayı var mıdır?Twin primes are prime numbers that differ by 2. Are there an infinite number of twin prime numbers?
[102]
vixra:2305.0122 [
pdf]
Pythagorean Triples and the Binomial Formula
The article shows the possibility of compiling Pythagorean triples using the binomial formula and provides a Theorem that is an alternative proof of the infinity of Pythagorean triples and confirmation of the close connection of the Pythagorean Theorem with the binomial formula.
[103]
vixra:2304.0222 [
pdf]
The Asymptotic Squeeze Principle and the Binary Goldbach Conjecture
In this paper, we prove the special squeeze principle for all sufficiently large $nin 2mathbb{N}$. This provides an alternative proof for the asymptotic version of the binary Goldbach conjecture in cite{agama2022asymptotic}.
[104]
vixra:2304.0218 [
pdf]
New Prime Number Theory
This paper introduces a novel approach to estimating the distribu- tion of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. Application of this methodology to infinite sums and nth terms, and propose several ways of defining the nth term of a prime number. By using the Ramanujan infinite series of natural numbers, I am able to derive an infinite series of prime numbers value . Overall, this work represents a significant contribution to the field of prime number theory and sheds new light on the relationship between prime numbers and other mathematical concepts.
[105]
vixra:2304.0209 [
pdf]
Complex Circles of Partition and the Expansion Principles
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the expansion principles as central tools for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.
[106]
vixra:2304.0192 [
pdf]
Irrationality of Pi Using Just Derivatives
The quest for an irrationality of pi proof that can be incorporated into an analysis (or a calculus) course is still extant. Ideally a proof would be well motivated and use in an interesting way the topics of such a course. In particular $e^{pi i}$ should be used and the more easily algebraic of derivatives and integrals -- i.e. derivatives. A further worthy goal is to use techniques that anticipate those needed for other irrationality and, maybe even, transcendence proofs. We claim to have found a candidate proof.
[107]
vixra:2304.0181 [
pdf]
The Randomness in the Prime Numbers
The prime numbers have very irregular pattern. The problem of finding pattern in the prime numbers is the long-standing open problem in mathematics. In this paper, we try to solve the problem axiomatically. And we propose some natural properties of prime numbers.
[108]
vixra:2304.0166 [
pdf]
Proof of the Triple and Twin Prime Conjectures Using the Sindaram Sieve Method
Yitang Zhang proved in 2013 that there are infinitely many pairs of prime numbers differing by 70 million, it has been proved now that there are infinitely many pairs of prime numbers differing by 246. In this paper, we use the sievemethod invented by Snndaram in 1934 to find out the solution of triple prime numbers and twin prime numbers, and find the general solution formula of the subset, i.e, an1 + b which is result of each subset, such as 3n + 1, 5n + 2, 7n + 3, 9n + 4, 11n+ 5, 13n+ 6, 15n+ 7, 17n+ 8, · · · in 2mn+n+m, modulo x respectively (x ≤ 3 takes prime). This general solution formula is used to prove the triple prime conjecture and the twin prime conjecture.
[109]
vixra:2304.0049 [
pdf]
On Goldbach Conjecture and Twin Prime Conjecture Part One: History, Development and Doubt
In this paper, we introduce Goldbach Conjecture and Twin Prime Conjecture: history, development, public dissemination in China, and propose doubt about the effectiveness of Analytical Number Theory
[110]
vixra:2304.0001 [
pdf]
A Proof of the Twin Prime Conjecture
It is well known to mathematicians, that there is an infinite number of primes as proven via simple logic by Euclid in the 4th Century BC1,2 and confirmed by Leonhard Euler in 17373. In 1846 French mathematician Alphonse de Polignac4 proposed that any even number can be expressed in infinite ways as the difference between two consecutive primes, since when or perhaps possibly even before that all the way back to Euclid, mathematicians have been trying to prove that there is an infinite number of TWIN PRIMES. In this paper a relatively simple proof is presented, that there is indeed an infinity of TWIN PRIMES based on a new approach without any assumptions.
[111]
vixra:2303.0102 [
pdf]
The Riemann Hypothesis Is True: The End of the Mystery
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) s=sigma+it of the zeta function, defined by: zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,for Re(s)>1 have real part} sigma= 1/2. In this note, I give the proof that sigma= 1/2 using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet eta function
[112]
vixra:2303.0091 [
pdf]
Collatz Conjecture
The paper analyzes the number of zeros in the binary representation of a natural number. The analysis is carried out using the concept of the fractional part of a number, which naturally arises when finding a binary representation. This idea relies on the fundamental property of the Riemann zeta function, which is constructed using the fractional part of a number. Understanding that the ratio of the fractional and integer parts, by analogy with the Riemann zeta function, expresses the deep laws of numbers, will explain the essence of this work. For the Syracuse sequence of numbers that appears in the Collatz conjecture, we use a binary representation that allows us to obtain a uniform estimate for all terms of the series, and this estimate depends only on the initial term of the Syracuse sequence. This estimate immediately leads to the solution of the Collatz conjecture
[113]
vixra:2302.0148 [
pdf]
General Relativity Theory of Numbers
In this paper we show that thorough understanding of numbers is possible only if we present them as value in relation to the certain reference measure. Commonly, we use number 1 as a reference measure, however, it does not have to always be 1, it can be any other number. To fully understand the meaning of numbers, we have to maintain their natural form which is a quotient of a value to a reference measure. Only by keeping this form we can do mathematics properly and appreciate its natural beauty.
[114]
vixra:2301.0082 [
pdf]
On Solutions to Erdos-Straus Equation Over Certain Integer Powers
We apply the notion of the textbf{olloid} to show that the ErdH{o}s-Straus equation $$frac{4}{n^{2^l}}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solutions for all $lgeq 1$ provided the equation $$frac{4}{n}=frac{1}{x}+frac{1}{y}+frac{1}{z}$$ has solution for a fixed $n>4$.
[115]
vixra:2301.0011 [
pdf]
Abridged Riemann's Last Theorem
The central idea of this article is to introduce and prove a special form of the zeta function as proof of Riemann’s last theorem. The newly proposed zeta function contains two sub-functions, namely f1(b,s) and f2(b,s) . The unique property of zeta(s)=f1(b,s)-f2(b,s) is that as tends toward infinity, the equality zeta(s)=zeta(1-s) is transformed into an exponential expression for the zeros of the zeta function. At the limiting point, we simply deduce that the exponential equality is satisfied if and only if real(s)=1/2. Consequently, we conclude that the zeta function cannot be zero if real(s)=1/2, hence proving Riemann’s last theorem.
[116]
vixra:2212.0209 [
pdf]
A Complete Proof of the Conjecture ca Complete Proof of the Conjecture C Smaller Than Rad^{1.63}(abc)
In this paper, we consider the $abc$ conjecture, we will give the proof that the conjecture $c smaller than rad^{1.63}(abc)$ is true. It constitutes the key to resolve the $abc$ conjecture.
[117]
vixra:2212.0162 [
pdf]
On the Gauss Circle Problem
Using the method of compression, we prove an inequality related to the Gauss circle problem. Let $mathcal{N}_r$ denotes the number of integral points in a circle of radius $r>0$, then we have $$2r^2bigg(1+frac{1}{4}sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r}) leq mathcal{N}_r leq 8r^{2}bigg(1+sum limits_{1leq kleq lfloor frac{log r}{log 2}floor}frac{1}{2^{2k-2}}bigg)+O(frac{r}{log r})$$ for all $r>1$. This implies that the error function $E(r)$ of the counting function $mathcal{N}_rll r^{1-epsilon}$ for any $epsilon>0.$
[118]
vixra:2212.0141 [
pdf]
On the Finiteness of Sequences of Even Squarefree Fibonacci Numbers
Let 2p1p2 . . . pk−1 be an even squarefree Fibonacci number with k distinct prime factors. For each positive k, such numbers form an integersequence. We conjecture that each such sequence has only a finite number of terms. In particular, the factorization data for the first 1000 Fibonacci numbers suggests that there is only one such term for k = 2, 5 for k = 3, and 8 for k = 4. We also renew attention to the fact that a proof that there are infinitely many squarefree Fibonacci numbers remains lacking. Some approachto proving this, emerging from our study, is suggested.
[119]
vixra:2212.0011 [
pdf]
A Proof of the Scholz Conjecture on Addition Chains
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequality $$iota(2^n-1)leq n-1+iota(n)$$ where $iota(n)$ denotes the length of the shortest addition chain producing $n$.
[120]
vixra:2211.0139 [
pdf]
A Refined Pothole Method and the Scholz Conjecture on Addition Chains
Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove the inequality $$iota(2^n-1)leq frac{3}{2}n-left lfloor frac{n-2}{2^{lfloor frac{log n}{log 2}-1floor+1}}ight floor-lfloor frac{log n}{log 2}-1floor +frac{1}{4}(1-(-1)^n)+iota(n)$$ where $lfloor cdot floor$ denotes the floor function and $iota(n)$ the shortest addition chain producing $n$.
[121]
vixra:2211.0120 [
pdf]
On the Irrationality of Riemann Zeta Functional Value at Odd Integers
In this article we provide a proof of the irrationality of ζ(2n+1) ∀n ∈ N.Also,in our attempt, we construct an upper bound to the Zeta values at odd integers.It is interesting to see how the irrationality of Zeta values at even positive integers mixed up with Dirichletsirrationality criterion and this bound accelerates our proof further,case by case.
[122]
vixra:2211.0095 [
pdf]
A Formula of the Dirichlet Character Sum
In this paper, We use the Fourier series expansion of realvariables function, We give a formula to calculate the Dirichlet charactersum, and four special examples are given
[123]
vixra:2211.0068 [
pdf]
A Note About The Determination of Integer Coordinates of Elliptic Curves - Part II, v1 -
In this paper, we give an elliptic curve $(E)$ given by the equation:y^2=f(x)=x^3+px+q with $p,qin Z$ not null simultaneous. We study the conditions verified by $(p,q)$ so that $exists ,(x,y) in Z^2$ the coordinates of a point of the elliptic curve $(E)$ given by the equation above. Key words: elliptic curves, integer points, solutions of degree three polynomial equations, solutions of Diophantine equations.
[124]
vixra:2211.0046 [
pdf]
On a Conjecture of Erdos on Additive Basis of Large Orders
Using the methods of multivariate circles of partition, we prove that for any additive base A [...] holds for sufficiently large values of k provided the counting function [...] is an increasing function for all k sufficiently large.
[125]
vixra:2210.0119 [
pdf]
On the Distribution of Perfect Numbers and Related Sequences via the Notion of the Disc
In this paper we investigate some properties of perfect numbers and associated sequences using the notion of the disc induced by the sum-of-the-divisor function $sigma$. We reveal an important relationship between perfect numbers and abundant numbers.
[126]
vixra:2210.0054 [
pdf]
Complex Circles of Partition and the Asymptotic Lemoine Conjecture
Using the methods of the complex circles of partition (cCoPs), we study textit{interior} and textit{exterior} points of such structures in the complex plane. With similarities to textit{quotient groups} inside of the group theory we define textit{quotient cCoPs}. With it we can prove an asymptotic version of the textbf{Lemoine Conjecture}.
[127]
vixra:2210.0029 [
pdf]
Olutions to the Exponential Diophantine 1 + P_1^x + P_2^y + P_3^z = W^2 for Distinct Primes P_1, P_2, P_3.
We list non-negative integer solutions (x,y,z,w) to the 5-term exponential diophantine equation 1 +p_1^x +p_2^y + p_3^z = w^2 for three distinct primes p_1 < p_2 < p_3 <= 113 obtained by exhaustive search in the exponents x, y, z <= 60.
[128]
vixra:2210.0013 [
pdf]
The Asymptotic Binary Goldbach and Lemoine Conjecture
In this paper we use the former of the authors developed theory of circles of partition to investigate possibilities to prove the binary Goldbach as well as the Lemoine conjecture. We state the squeeze principle and its consequences if the set of all odd prime numbers is the base set. With this tool we can prove asymptotic versions of the binary Goldbach as well as the Lemoine conjecture.
[129]
vixra:2209.0167 [
pdf]
A Problem on Sums of Powers
We pose a problem which is motivated by Newton's identity on sums of powers. We prove two special cases using algebraic manipulations. The method used is inefficient to prove all cases.
[130]
vixra:2209.0107 [
pdf]
Locally Uniform Approximations and Riemann Hypotheses (Fourth Revised)
This paper offers a breakthrough in proving the veracity of original Riemann hypothesis, and extends the validity of its method to include the cases of the Dedekind zeta functions, the Hecke L-functions hence the Artin L-functions, and the Selberg class.First we parametrize the Riemann surface $mathbf{S}$ of $log$-function, with which we first shrink the scale of each chosen parameter for which it depends on the chosen natural number $Q_{N_{0}}$ which is a chosen common multiple of all the denominators which are derived from a pre-set choice of rational numbers which approximate the values $log(k+1)$ with the integers $k$ in $0leq kleq N$.Then in (1.7) we define the mapping $-Q_{N_{0}}log(.)$ to pull the truncated Dirichlet $eta$-function $f_{N}(s)$ back to be re-defined on $mathbf{S}$, after that we shrink all the points to have their absolute values are all less than $1$ and closer to $1$. We apply the Euler transformation to the alternative series of Dirichlet $eta$-functions $f(s)$ which are defined in (1.4), then we build up the locally uniform approximation of Theorem 4.7 for $f(s)$ which are established on any given compact subset contained in the right half complex plane.In the second part we define the functions $phi(s)$ which are formulated in (6.1) then by specific property of the functions $phi(s)$, we have the similar asymptotics Theorem 6.5 as those of Theorem 4.7 to obtain the result of Theorem 6.8.And with the locally uniform estimation Lemma 5.10, finally in Theorem 5.9 and Theorem 6.9 we employ Theorem 5.8 and Theorem 6.8 to solve problems of Riemann hypothesis for the Dedekind zeta functions, the Hecke L-functions, the Artin L-functions, and the Selberg class for which all of their nontrivial zeros are contained in the vertical line $Re(s)=1/2$.Finally for the $gamma(s)$-factor of each Dirichlet series $D(s)$ which is formulated in (1.4), then by Theorem 6.9 it has neither zeros nor poles contained in the critical strip $0<Re(s)<1$ and the non-existence of Siegel's zeros for such Dirichlet series $D(s)$ is confirmed.
[131]
vixra:2208.0131 [
pdf]
On the General Erdh{o}s-Moser Equation Via the Notion of Olloids
We introduce and develop the notion of the textbf{olloid}. We apply this notion to study a variant and a generalized version of the ErdH{o}s-Moser equation under some special local condition.
[132]
vixra:2208.0041 [
pdf]
General Base Decimals with the P-Series of Calculus Shows All Zeta(n) Irrational
We give a new approach to the question of whether or not all greater than one, integer arguments of Zeta are irrational. Currently only Zeta(2n) and Zeta(3) are known to be irrational. We show that using the denominators of the terms of Zeta(n)-1=z_n as decimal bases gives all rational numbers in (0,1) as single decimals, property one. We also show the partial sums of z_n are not given by such single digits so using the denominators of the partial sum's terms as number bases, property two. Next, using integrals for the p-series contracting upper and lower bounds for partial sum remainders of z_n are generated. Assuming z_n is rational, it is expressible as a single decimal using the denominator of a term of z_n (property one) and eventually these bounds will consist of infinite decimals (property two) with their first decimal equal to this single decimal. But as no single decimal can be between two infinite decimals with the same first digit a contradiction is derived and all z_n are proven irrational.
[133]
vixra:2208.0025 [
pdf]
Fermat's Last Theorem: A Proof by Contradiction
In this paper I oer an algebraic proof by contradiction of Fermat's Last Theorem. Using an alternative to the standard binomial expansion, (a+b)n = an + b Pn i=1 ani(a + b)i1, a and b nonzero integers, n a positive integer, I showthat a simple rewrite of the equation stating the theorem, Ap + (A + b)p = (2A + b c)p; A; b and c positive integers, entails the contradiction of two positive integers that sum to less than zero,(2f + g)(f + g)(f + g + b) Xp2 i=1 (2f + g)p2i(3f + 2g + b)i1 + (f + b)(f + g)(3f + 2g + b)p2 + fb(3f + 2g + b)p2 < 0; f and g positive integers. This contradiction shows that the rewrite has nonon-trivial positive integer solutions and proves Fermat's Last Theorem.
[134]
vixra:2208.0016 [
pdf]
A Lower Bound for Length of Addition Chains
In this paper we show that the shortest length $iota(n)$ of addition chains producing numbers of the form $2^n-1$ satisfies the lower bound $$iota(2^n-1)geq n+lfloor frac{log (n-1)}{log 2}floor$$ where $lfloor cdot floor$ denotes the floor function.
[135]
vixra:2208.0007 [
pdf]
A Simple Proof that Goldbach's Conjecture is True
A induction proof shows Goldbach's conjecture is correct. It is as simple as can be imagined. A table consisting of two rows is used. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. For the base case of k=5 suppose that primes on the lower row always map to composites on the top and that this results in too many composites on the top. This is true for this base case. Suppose it is true for k=n, then the shifts and additions necessary for the k=n+1 case maintain this property of too many composites on top. The contrapositive is that there exists a prime on the bottom that maps to a prime on top and Goldbach is established: the sum of these two primes is 2(n+1).
[136]
vixra:2207.0119 [
pdf]
On a Certain Inequality on Addition Chains
In this paper we prove that there exists an addition chain producing 2^n-1 of length delta(2^n-1) satisfying the inequality delta(2^n-1)leq 2n-1-2left lfloor frac{n-1}{2^{lfloor frac{log n}{log 2}floor}}ight floor+lfloor frac{log n}{log 2}flooronumber where lfloor cdot floor denotes the floor function.
[137]
vixra:2207.0086 [
pdf]
Multidimensional Numbers
A multidimensional number will not be viewed as a single real scalar value, rather,as a set of scalar values, each associated with a dimension. This gives rise to variations of"complex numbers", and consequently, Euler’s formula. The properties of complex numbers,such as the product of magnitudes being equal to the magnitude of the products, may also beapplicable in the case of multidimensional numbers, depending on how they are constructed.Although similar ideas exist, such as a hypercomplex number, the differences will be discussed.
[138]
vixra:2207.0060 [
pdf]
The prime Number Theorem and Prime Gaps
Let there exists m > 0 such that gn = O((logpn)m), then∀k > 0, ∃M ∈ N s.t. n ≥ M ⇒ gn := pn+1 − pn < pknwhere pn is nth prime number, O is big O notation, log is natural logarithm.This lead to a corollary for Andrica conjecture, Oppermann conjecture.
[139]
vixra:2207.0013 [
pdf]
Proofs of Four Conjectures in Number Theory: Beal's Conjecture, Riemann Hypothesis, The abc and c Smaller Than R^{1.63} Conjectures
This monograph presents the proofs of 4 important conjectures in the field of the number theory, namely:- Beal's conjecture. - The Riemann Hypothesis.- The c smaller than R^{1.63} conjecture is true. - The abc conjecture is true. We give the details of the different proofs.
[140]
vixra:2207.0011 [
pdf]
Proof of 16 Formulas Barnes Function
I have already published several months ago in the papers "Values of Barnes Function" and "Another Values of Barnes Function and Formulas" in total 16 conjectural formulas that I find with unsualmethods.So, in this article, I write the proof of 16 formulas.
[141]
vixra:2206.0168 [
pdf]
Bernoulli Sums of Powers, Euleru2013maclaurin Formula and Proof that Riemann Hypothesis is True
On 1859, the german mathematician Georg Friedrich Bernhard Riemann made one of his most famouspublications "On the Number of Prime Numbers less than a Given Quantity" when he was developing hisexplicit formula to give an exact number of primes less than a given number x, in which he conjectured that"all non-trivial zeros of the zeta function have a real part equal to 1/2 ". Riemann was sure of his statement,but he could not prove it, remaining as one of the most important hypotheses unproven for 163 years.In this paper, we have to prove that the Riemann Hypothesis is true, based on the Bernoullipower sum, the Euleru2013Maclaurin formula and its relation with the Riemann Zeta function.
[142]
vixra:2206.0164 [
pdf]
What Makes Goldbach's Conjecture Correct
A direct proof shows Goldbach's conjecture is correct. It is as simple as can be imagined. A table consisting of two rows is used. The lower row counts from 0 to any n and and the top row counts down from 2n to n. All columns will have all numbers that add to 2n. Using a sieve, all composites are crossed out and only columns with primes are left. Without loss of generality, an example shows that primes, ones that sum to 2n will always be left in such columns.
[143]
vixra:2206.0075 [
pdf]
A Formula for the Function π(x) to Count the Number of Primes Exactly if 25 ≤ X ≤ 1572 with Python Code to Test it v. 4.0
This paper shows a very elementary way of counting the number of primes under a given number with total accuracy. Is the function π(x) if 25 ≤ x ≤ 1572.
[144]
vixra:2206.0051 [
pdf]
Proof of Riemann Hypothesis
This paper is a trial to prove Riemann hypothesis accordingto the following process.1. We make one identity regarding x from one equation that gives Riemannzeta function ζ(s) analytic continuation and 2 formulas (1/2+a±bi, 1/2−a ± bi) that show non-trivial zero point of ζ(s).2. We find that the above identity holds only at a = 0.3. Therefore non-trivial zero points of ζ(s) must be 1/2 ± bi because a cannothave any value but zero.
[145]
vixra:2206.0018 [
pdf]
On Riemann Hypothesis
A line of study of the Riemann Hypothesis is proposed, based on a comparison with Weil zeros and a categorification of the duality between Riemann zeros and prime numbers. The three case of coefficients, complex, p-adic and finite fields are also related.
[146]
vixra:2205.0155 [
pdf]
Algorithm for Finding the Nth Root of Modulo P
For { p-1 = q^L*m ( ∤ q^x ∨ | q^x (x larger than L))}, it is the deterministic algorithm. The previously created calculation method was for a single prime number, but a method to calculate multiple primes has been added. The original calculation method has also been partially modified. To find the nth root, we need to factoer n into prime factors. In some case, primitive roots are needed. If you don't know these, use the Tonelli-Shanks algorithm.
[147]
vixra:2205.0151 [
pdf]
On the Infinitude of Cousin Primes
In this paper we prove that there infinitely many cousin primes by deducing the lower bound \begin{align} \sum \limits_{\substack{p\leq x\\p,p+4\in \mathbb{P}\setminus \{2\}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align}where $\mathcal{C}:=\mathcal{C}(4)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular it follows that \begin{align} \sum \limits_{p,p+4\in \mathbb{P}\setminus \{2\}}1=\infty\nonumber \end{align}by taking $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$.
[148]
vixra:2205.0132 [
pdf]
Comprehending the Euler-Riemann Zeta Function and a Proof of the Riemann Hypothesis
This paper will prove that the Riemann Hypothesis is true., based on the following statements: -The resulting value of the Euler-Riemann zeta function ζ(k) is the center of a spiral on the complex plane, where k ∈ C. -The center of this spiral when ζ(k) = 0, coincides with the origin of coordinates of the complex plane. -There exists a function related to this spiral, obtained from Bernoulli's sum of powers, which allows to calculate the zeta funtion.
[149]
vixra:2205.0106 [
pdf]
On a Variant of Brocard's Problem Via the Diagonalization Method
In this paper we introduce and develop the method of diagonalization of functions $f:mathbb{N}longrightarrow mathbb{R}$. We apply this method to show that the equations of the form $Gamma_r(n)+k=m^2$ has a finite number of solutions $nin mathbb{N}$ with $n>r$ for any fixed $k,rin mathbb{N}$, where $Gamma_r(n)=n(n-1)cdots (n-r)$ denotes the $r^{th}$ truncated Gamma function.
[150]
vixra:2205.0084 [
pdf]
On the Length of Addition Chains Producing $2^n-1$
Let $\delta(n)$ denotes the length of an addition chain producing $n$. In this paper we prove that the exists an addition chain producing $2^n-1$ whose length satisfies the inequality $$\delta(2^n-1)\lesssim n-1+\iota(n)+\frac{n}{\log n}+1.3\log n\int \limits_{2}^{\frac{n-1}{2}}\frac{dt}{\log^3t}+\xi(n)$$ where $\xi:\mathbb{N}\longrightarrow \mathbb{R}$. As a consequence, we obtain the inequality $$\iota(2^n-1)\lesssim n-1+\iota(n)+\frac{n}{\log n}+1.3\log n\int \limits_{2}^{\frac{n-1}{2}}\frac{dt}{\log^3t}+\xi(n)$$ where $\iota(n)$ denotes the length of the shortest addition chains producing $n$.
[151]
vixra:2205.0028 [
pdf]
On Addition Chains of Fixed Degree
In this paper we extend the so-called notion of addition chains and prove an analogue of Scholz's conjecture on this chain. In particular, we obtain the inequality $$\iota^{\lfloor \frac{n-1}{2}\rfloor}(2^n-1)\leq n+\iota(n)$$ where $\iota(n)$ and $\iota^{\lfloor \frac{n-1}{2}\rfloor}(n)$ denotes the length of the shortest addition chain and the shortest addition chain of degree $\lfloor \frac{n-1}{2}\rfloor$, respectively, producing $n$.
[152]
vixra:2205.0008 [
pdf]
Generating and Deconstructing Prime Numbers
Prime numbers have a rich structure, when viewed as sizes of finite fields. Iteration of an analysis as Klein geometry yields their deconstruction into simpler primes: the POSet structure. Reversing the process is Euclid's trick of generating new primes. A generalization of this is used by McCanney to cover the set of primes away from primorials as centers. This fast algorithm has a ``propagation'' flavor. Generating primes in this manner is also related with Goldbach's Conjecture.
[153]
vixra:2204.0172 [
pdf]
Assuming c Less Than Rad*2(abc), The abc Conjecture Is False
In this paper, we consider the abc conjecture. Assuming that c<rad^2(abc) is true, we give anelementary proof that the abc conjecture is false using an equivalent statement.
[154]
vixra:2204.0143 [
pdf]
The Series of Reciprocals of The Primes Diverges
This paper gives a detailed proof of Euler's theorem, which is the divergence of a series of reciprocals of the primes. The key idea of the proof is to assume the series converges and then complete the proof by contradiction.
[155]
vixra:2204.0105 [
pdf]
On Prime Numbers and Riemann Zeros
Intuitively, prime numbers of ``Number systems'' (rings) are the building blocks of their elements. We start from natural numbers and Gaussian integers to explain more general frameworks, like the structure theorem for finitely generated Abelian groups. We end with a 1 million dollar puzzle, the Riemann Hypothesis, and point to the fact that prime numbers are dual to the Riemann zeros. Some easy references are provided.
[156]
vixra:2204.0054 [
pdf]
Algorithm for Finding Q^k-th Root of a
Description of the algorithm for finding the q^k-th root of a. There is no basic difference in the calculation method created in the previous version. Some additions and changes have been made.
[157]
vixra:2204.0050 [
pdf]
Assuming C Less Than Rad^2(abc) and the Beal's Conjecture Hold, Then the Abc Conjecture is False
In this paper, assuming that the conjecture c<rad^2(abc) and Beal's Conjecture hold, I give, using elementary logic, the proof that the abc conjecture is false.
[158]
vixra:2204.0033 [
pdf]
{The Irrationality of Odd and Even Zeta Values
We show that using the denominators of the terms of $\zeta(n)-1=z_n$ as decimal bases gives all rational numbers in (0,1) as single decimals. We also show the partial sums of $z_n$ are not given by such single digits using the partial sum's terms. These two properties yield a proof that $z_n$ is irrational. As partials require denominators exceeding the denominators of their terms, possible single decimal convergence points are, using properties of decimal expansions, systematically eliminated.
[159]
vixra:2204.0031 [
pdf]
Improvement of Prime Number Theorem using the Multi-Point Summation Method
I propose a new approximate asymptotic formula for Prime number theorem. The new formula is derived by the multi-point summation method. It has additional term expressed with elementary function and gives better estimate of the prime-counting function from small value to big value. It also satisfies asymptotic formula with n -> ∞ limit.
[160]
vixra:2203.0183 [
pdf]
Collatz Conjecture: An Order Machine
Collatz conjecture (3n+1 problem) is an application of Cantor's isomorphism theorem (Cantor-Bernstein) under recursion. The set of 3n+1 for all odd positive integers n, is an order isomorphism for (odd X, 3X+1). The other (odd X, 3X+1) linear order has been discovered as a bijective order-embedding, with values congruent to powers of four. This is demonstrated using a binomial series as a set rule, then showing the isomorphic structure, mapping, and cardinality of those sets. Collatz conjecture is representative of an order machine for congruence to powers of two. If an initial value is not congruent to a power of two, then the iterative program operates the (odd X, 3X+1) order isomorphism until an embedded value is attained. Since this value is a power of four, repeated division by two tends the sequence to one. Because this same process occurs, regardless of the initial choice for a positive integer, Collatz conjecture is true.
[161]
vixra:2203.0170 [
pdf]
The Spanning Method and the Lehmer Totient Problem
In this paper we introduce and develop the notion of spanning of integers along functions $f:mathbb{N}longrightarrow mathbb{R}$. We apply this method to a class of problems requiring to determine if the equations of the form $tf(n)=n-k$ has a solution $nin mathbb{N}$ for a fixed $kin mathbb{N}$ and some $tin mathbb{N}$. In particular, we show that begin{align}# {nleq s~|~tvarphi(n)+1=n,~mathbf{for~some}~tin mathbb{N}}geq frac{s}{2log s}prod limits_{p | lfloor sfloor }(1-frac{1}{p})^{-1}-frac{3}{2}e^{gamma}onumberend{align}for $sgeq s_o$, where $varphi$ is the Euler totient function and $gamma=0.5772cdots$ is the Euler-Macheroni constant
[162]
vixra:2203.0128 [
pdf]
Every Sufficiently Large Even Number Is the Sum of Two Primes
The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha > 4$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 4$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.
[163]
vixra:2203.0117 [
pdf]
Corrections about V.S. Adamchik's Papers
I study several papers of V.S. Adamchik and I find several mistakes about integrals and the Melzak's product.In the same time, I give more general formulas of three integrals.
[164]
vixra:2203.0092 [
pdf]
On the Scholz Conjecture
In this paper we prove an inequality relating the length of addition chains producing number of the form $2^n-1$ to the length of their shortest addition chain producing their exponents. In particular, we obtain the inequality $$\delta(2^n-1)\leq n-1+\iota(n)+G(n)$$ where $\delta(n)$ and $\iota(n)$ denotes the length of an addition chain and the shortest addition chain producing $n$, respectively, with $G:\mathbb{N}\longrightarrow \mathbb{R}$.
[165]
vixra:2203.0050 [
pdf]
A Proof that Zeta(n >= 2) is Irrational
We show that using the denominators of the terms of Zeta(n)-1=z_n as decimal bases gives all rational numbers in (0,1) as single decimals. We also show the partial sums of z_n are not given by such single digits using the partial sum's terms. These two properties yield a proof that z_n is irrational.
[166]
vixra:2202.0037 [
pdf]
Proof of Fermat's Last Theorem by means of Elementary Probability Theory
In this work, we introduce the concept of Fermat’s Urn, an urn containing three types of marbles, and such that it holds a peculiar constraint therein: The probability to get at least one marble of a given type (while performing multiple independent drawings) is equal to the probability not to get any marble of another type. Further, we discuss a list of implicit hypotheses related to Fermat's Equation, which would allow us to interpret this equation exactly as the mentioned constraint in Fermat's Urn. Then, we study the properties of this constraint in relation with the capability to distinguish the types of marbles within the urn, namely in case of the event ''to get at least one marble of each type''. Eventually, on the basis of a simple theorem related to this event, we prove that Fermat's Equation and Fermat's Urn may share those properties only if we perform at most two drawings from the urn. This result reflects then in the solution of Fermat's Equation.
[167]
vixra:2202.0026 [
pdf]
A Formula for the Function π(x) to Count the Number of Primes Exactly if 25 ≤ X ≤ 1572 with Python Code to Test it v. 4.0
This paper shows a very elementary way of counting the number of primes under a given number with total accuracy. Is the function π(x) if 25 ≤ x ≤ 1572.
[168]
vixra:2201.0215 [
pdf]
The Gaps Between Primes
It is proved that· For any positive integer d, there are infinitely many prime gaps of size 2d.· Every even number greater than 2 is the sum of two prime numbers.Our method from the analysis of distribution density of pseudo primes in specific set is to transform them into upper bound problem of the maximum gaps between overlapping pseudo primes, then the two are essentially the same problem.
[169]
vixra:2201.0194 [
pdf]
Another Values of Barnes Function and Formulas
In this paper,I study values of Barnes G function as G(k/8) and G(k/12) with Wallis product as applications.And I write several formulas, so we can evaluate elementary values.
[170]
vixra:2201.0048 [
pdf]
Generalized Cannonball Problem
The cannonball problem asks which numbers are both square and square pyramidal. In this paper I consider the cannonball problem for other $r$-regular polygons. I carried out a computer search and found a total of $858$ solutions for polygons $3\le r\le10^5$. By using elliptic curves I also found that there are no solutions for $r=5$ (pentagon), $r=7$ (heptagon), and $r=9$ (enneagon).
[171]
vixra:2112.0161 [
pdf]
The Diagonalization Method and Brocard's Problem
In this paper we introduce and develop the method of diagonalization of functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to a class of problems requiring to determine if the equations of the form $f(n)+k=m^2$ has a finite number of solutions $n\in \mathbb{N}$ for a fixed $k\in \mathbb{N}$.
[172]
vixra:2112.0153 [
pdf]
Singly and Doubly Even Multiples of 6 and Statistical Biases in the Distribution of Primes
Computer experiments show that singly even multiples of 6 sur-rounded by prime pairs exhibit a larger ratio of nonsquarefree to squarefree multiples than generic singly even multiples of 6, a bias of ca 10.6% measured against the expected value. The same bias occurs for isolated primes next to singly even multiples of 6; here the deviation from the expected value is ca 3.3% of this value. The expected value of the ratio of singly even to doubly even nonsquarefree multiples of 6 also differs from values found experimentally for prime pairs centered on such multiples or isolated primes next to them. For pairs, this ratio exceeds its unbiased value by ca 6.2%, for isolated primes by ca 2.0%. The values cited are for the first 10^10 primes, the largest range we investigated. This paper broadens our recent study of a newly found bias in the distribution of primes by examining singly and doubly even multiples of 6. In particular, it shows that for primes centered on or next to singly even multiples of 6, the statistical biases in question are more pronounced than in the general case studied by us before.
[173]
vixra:2112.0145 [
pdf]
Riemann’s Last Theorem
The central idea of this article is to introduce and prove a special form of the zeta function as proof of Riemann’s last theorem. The newly proposed zeta function contains two sub functions, namely f1(b,s) and f2(b,s) . The unique property of zeta(s)=f1(b,s)-f2(b,s) is that as tends toward infinity the equality zeta(s)=zeta(1-s) is transformed into an exponential expression for the zeros of the zeta function. At the limiting point, we simply deduce that the exponential equality is satisfied if and only if real(s)=1/2 . Consequently, we conclude that the zeta function cannot be zero if real(s)=1/2 , hence proving Riemann’s last theorem.
[174]
vixra:2112.0081 [
pdf]
On the Last Numbers of Positive Integers
In this note, we are interested in the last numbers of positive integers; for example, for 20211206, the last number is 6, typically we note that for any positive integer a, the last numbers of a5 and a are the same.
[175]
vixra:2112.0067 [
pdf]
Tentatives For Obtaining The Proof of The Riemann Hypothesis
This report presents a collection of some tentatives to obtain a final proof of the Riemann Hypothesis. The last paper of the report is submitted to a mathematical journal for review.
[176]
vixra:2112.0061 [
pdf]
On the General Erd\h{o}s-Tur\'{a}n Additive Base Conjecture
In this paper we introduce a multivariate version of circles of partition introduced and studied in \cite{CoP}. As an application we prove a weaker general version of the Erd\H{o}s-Tur\'{a}n additive base conjecture. The actual Erd\H{o}s-Tur\'{a}n additive base conjecture follows from this general version as a consequence.
[177]
vixra:2112.0027 [
pdf]
The Binary Goldbach Conjecture Via the Notion of Signature
In this paper we prove the binary Goldbach conjecture. By exploiting the language of circles of partition, we show that for all sufficiently large $n\in 2\mathbb{N}$ \begin{align} \# \left \{p+q=n|~p,q\in \mathbb{P}\right \}>0.\nonumber \end{align}This proves that every sufficiently large even number can be written as the sum of two prime numbers.
[178]
vixra:2112.0022 [
pdf]
A New Relation Between Lerch's $\Phi$ and the Hurwitz Zeta
A new relation between the Lerch's transcendent, $\Phi$, and the Hurwitz zeta, $\zeta(k,b)$, at the positive integers is introduced. It is derived simply by inverting the relation presented in the precursor paper with one of two approaches (its generating function or the binomial theorem). This enables one to go from Lerch as a function of Hurwitz zetas (of different orders), to Hurwitz as a function of Lerches. A special case of this new functional equation is a relation between the Riemann's zeta function and the polylogarithm.
[179]
vixra:2111.0168 [
pdf]
The Tower Function and Applications
In this paper we study an extension of the Euler totient function to the rationals and explore some applications. In particular, we show that \begin{align} \# \{\frac{m}{n}\leq \frac{a}{b}~|~m\leq a,~n\leq b,~\gcd(m,a)=\gcd(n,b)=1,~\gcd(n,a)>1\nonumber \\~\vee~\gcd(m,b)>1~\vee ~\gcd(m,n)>1\}=\sum \limits_{\substack{\frac{m}{n}\leq \frac{a}{b}\\mn\leq ab\\m>a,n\leq b~\vee~m\leq a,n>b~\vee~\gcd(m,n)>1\\ \gcd(mn,ab)=1}}1\nonumber \end{align} provided $\gcd(a,b)=1$.
[180]
vixra:2111.0132 [
pdf]
Proof of Riemann Hypothesis (3)
This paper is a trial to prove Riemann hypothesis according to the following process. 1. We make (N+1)/2 infinite series from one equation that gives ζ(s) analytic continuation and 2 formulas (1/2+a+bi, 1/2−a−bi) that show non-trivial zero point of ζ(s). (N = 1, 3, 5, 7, · · · · · · ) 2. We find that a cannot have any value but zero from the above infinite series by performing N → ∞. 3. Therefore non-trivial zero point of ζ(s) must be 1/2 ± bi.
[181]
vixra:2111.0130 [
pdf]
An Exact Formula for the Prime Counting Function
This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of a neat power series for the prime counting function, $\pi(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which may provide evidence for the Riemann hypothesis, and enabled the creation of a formula for $\pi(x)$ in the first place), and the realization that sums of divisors and the M\"{o}bius function are particular cases of a more general concept. One of its conclusions is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.
[182]
vixra:2111.0129 [
pdf]
Lerch's $\phi$ and the Polylogarithm at the Positive Integers
We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^k\zeta(k)b^{k-1}$). Finally, we use the found Fourier series formulae to obtain the values of the Lerch transcendent function, $\Phi(e^m,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, at the positive integers $k$.
[183]
vixra:2111.0128 [
pdf]
Lerch's $\phi$ and the Polylogarithm at the Negative Integers
At the negative integers, there is a simple relation between the Lerch $\Phi$ function and the polylogarithm. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Starting from that formula, we can deduce a simple closed formula for the Lerch $\Phi$ function at the negative integers, where the Stirling numbers are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $\Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we present a new formula for the Hurwitz zeta function at the positive integers using this novelty.
[184]
vixra:2111.0127 [
pdf]
A Reformulation of the Riemann Hypothesis
We present some novelties on the Riemann zeta function. Using the analytic continuation we created for the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, we extend the zeta function from $\Re(k)>1$ to the complex half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.
[185]
vixra:2111.0094 [
pdf]
A Revisit to Lemoine's Conjecture
In this paper we prove Lemoine's conjecture. By exploiting the language of circles of partition, we show that for all sufficiently large $n\in 2\mathbb{N}+1$ \begin{align} \# \left \{p+2q=n|~p,q\in \mathbb{P}\right \}>0.\nonumber \end{align}This proves that every sufficiently large odd number can be written as the sum of a prime and a double of a prime.
[186]
vixra:2111.0070 [
pdf]
A New Method for the Cubic Polynomial Equation
I present a method to solve the general cubic polynomial equation based on six years of research that started back in 1985 when, in the fifth grade, I first learned of Bhaskara's formula for the quadratic equation. I was fascinated by Bhaskara's formula and naively thought I could replicate his method for the third degree equation, but only succeeded in 1990, after countless failed attempts. The solution involves a simple transformation to form a cube and which, by chance, happens to reduce the degree of the equation from three to two (which seems to be the case of all polynomial equations that admit solutions by means of radicals). I also talk about my experiences trying to communicate these results to mathematicians, both at home and abroad.
[187]
vixra:2110.0178 [
pdf]
On the Prime Distribution
In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots, A_i. \ldots $, such that the number of prime pairs ($p_i$ and $q_i $, $p_i-q_i = p-q $) in $A_i $ increases gradually, where $i>0$. So twin prime conjecture is true. We also prove that for any even integer $m>2700$, there exist more than 10 prime pairs $(p,q)$, such that $p+q=m$. Thus Goldbach conjecture is true.
[188]
vixra:2110.0120 [
pdf]
On Odd Perfect Numbers
In this note, we introduce the notion of the disc induced by an arithmetic function and apply this notion to the odd perfect number problem. We show that under certain special local condition an odd perfect number exists by exploiting this concept.
[189]
vixra:2110.0114 [
pdf]
Algorithm for Finding Q^k-th Root of a [q=prime A≡x^(q^k) (Mod p) ]
We have created a handy tool that allows you to calculate the q^k power root of a easily and quickly. However, the calculation may require a primitive root, and if the calculation requires a primitive root and you do not know the primitive root, please use the Tonelli-Shanks algorithm.
[190]
vixra:2110.0033 [
pdf]
Proof of Goldbach's Conjecture and Twin Prime Conjecture
In this paper, we prove Twin prime conjecture and Goldbach’s conjecture. We do this in three stages by turns; one is ‘Application Principle of Mathematical Induction’, another is ‘Proof of Twin prime conjecture’ and the other is ‘Proof of Goldbach’s conjecture’. These three proofs are interconnected, so they help prove it. Proofs of Twin prime conjecture and Goldbach’s conjecture are proved by Application Principle of Mathematical Induction. And Twin prime conjecture is based on Goldbach’s conjecture. So, we can get the result, Twin prime and Goldbach’s conjecture are true. The reason why we could get the result is that I use twin prime’s characteristic that difference is 2 and apply this with Application Principle of Mathematical Induction. If this is proved in this way, It implies that the problem can be proved in a new way of proof.
[191]
vixra:2109.0166 [
pdf]
Proof of the Riemann Hypothesis
In this article we will prove the problem equivalent to the Riemann Hypothesis developed by Luis-Báez in the article ``A sequential Riesz-like criterion for the Riemann hypothesis''.
[192]
vixra:2109.0161 [
pdf]
Is The Riemann Hypothesis True? Yes, It Is. v(4)
In 1859, Georg Friedrich Bernhard Riemann announced the following conjecture, called Riemann Hypothesis: The nontrivial roots (zeros) $s=sigma+it$ of the zeta function, defined by: $$zeta(s) = sum_{n=1}^{+infty}frac{1}{n^s},,mbox{for}quad Re(s)>1$$have real part $sigma= ds frac{1}{2}$.We give the proof that $sigma= frac{1}{2}$ using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet $eta$ function.
[193]
vixra:2108.0077 [
pdf]
Riemann Hypothesis Proof Using an Equivalent Criterion of Balazard, Saias and Yor
In this manuscript we denote a unit disc by $\mathbb{D}=\{z\in \mathbb{C} \mid |z|<1\}$ and a semi plane as\\ $\mathbb{P}=\{s\in\mathbb{C}\mid \Re(s)>\frac{1}{2}\}$. We denote, $\mathbb{R}_{\geq 0}=\{x\in \mathbb{R}\mid x\geq 0\}$ and $\mathbb{R}_{\geq 1}=\{x\in \mathbb{R}\mid x\geq 1\}$. Considering non negative real axis as a branch cut, we define a map from slit unit disc to the slit plane as $s:\mathbb{D}\setminus \mathbb{R}_{\geq 0}\to \mathbb{P}\setminus\mathbb{R}_{\geq 1}$ defined as $s(z)=\frac{1}{1-\sqrt{z}}$ which is proved to be one-one and onto. Next, we define a function $f(z)=(s-1)\zeta(s)$ where $s=s(z)$ and both $s(z)$ and $f(z)$ are proved to be analytic in $\mathbb{D}\setminus \mathbb{R}_{\geq 0}$. Next we prove that $s=s(z)$ is a conformal map. We also show that $f$ is continuous at $0$. Using Cauchy's residue theorem to a keyhole contour and Lebesgue's dominated convergence theorem along with Schwarz reflection principle, we prove that, $$\int_{-\infty}^\infty \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt=0$$ This settles the Riemann Hypothesis because this relation is an equivalent version of Riemann Hypothesis as proved by Balazard, Saias and Yor [1].
[194]
vixra:2108.0066 [
pdf]
Is Complex Number Theory Free from Contradiction
With simple basic mathematics it is possible to demonstrate a conflicting result in complex number theory using Euler’s identity, simple trigonometry and deMoivre’s formula for n=2.
[195]
vixra:2108.0065 [
pdf]
Redundant Primes In Lemoine’s Conjecture
Lemoine’s conjecture (LC), still unsolved, states that all positive odd integer ≥ 7 can be expressed as the sum of a prime and an even semiprime. But do we need all primes to satisfy this conjecture? This work is devoted to selection of must-have primes and formulation of stronger version of LC with reduced set of primes.
[196]
vixra:2108.0064 [
pdf]
On Fast Search of First Confirmation Of Goldbach’s Strong Conjecture
Goldbach strong conjecture states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). Hypothesis still remains open and is confirmed experimentally for bigger and bigger n. This work studies different approaches to finding the first confirmation of this conjecture in order to select the most effective confirmation method.
[197]
vixra:2108.0058 [
pdf]
On 6k ± 1 Primes in Goldbach Strong Conjecture
Goldbach strong conjecture, still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). Each prime p>3 can be expressed as 6k ± 1. This work is devoted to studies of 6k ± 1 primes in Goldbach partitions and enhanced Goldbach strong conjecture with the lesser of twin primes of form 6k − 1 used as a baseline.
[198]
vixra:2108.0057 [
pdf]
Redundant Primes In Goldbach Partitions
Goldbach Strong Conjecture (GSC), still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). But do we need all primes to satisfy this conjecture? This work is devoted to selection of must-have primes and formulation of stronger version of GSC with reduced set of primes.
[199]
vixra:2108.0056 [
pdf]
Goldbach Strong Conjecture Verification Using Prime Numbers
Goldbach strong conjecture, still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). We can also formulate it from the opposite perspective: from a set of prime numbers you may pick two primes, sum them, and you will be able to build every even number n>2. This work is devoted to studies on sum of two prime numbers.
[200]
vixra:2108.0055 [
pdf]
Studies on Twin Primes in Goldbach Partitions of Even Numbers
Goldbach strong conjecture states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). This work is devoted to studies on twin primes present in Goldbach partitions. Based on executed experiments original Goldbach conjecture has been extended to a form that all even integers n>4 can be expressed as the sum of twin prime and prime.
[201]
vixra:2108.0024 [
pdf]
On the Shortest Addition Chain of Numbers of Special Forms
In this paper we study the shortest addition chains of numbers of special forms. We obtain the crude inequality $$\iota(2^n-1)\leq n+1+G(n)$$ for some function $G:\mathbb{N}\longrightarrow \mathbb{N}$. In particular we obtain the weaker inequality $$\iota(2^n-1)\leq n+1+\left \lfloor \frac{n-2}{2}\right \rfloor$$ where $\iota(n)$ is the length of the shortest addition chain producing $n$.
[202]
vixra:2107.0137 [
pdf]
Acceptable Facts Point to Validity of Riemann Hypothesis
In this short note, I provide a proof for the Riemann Hypothesis. You are free not to get enlightened about that fact. But please pay respect to new dispositions of the Riemann Hypothesis and research methods in this note. I start with Dr.Zhu who was the first to show me that instead of the known 40 %, the maximum percentage of the zeroes of the Riemann zeta function belongs to the 1/2 critical line.
[203]
vixra:2107.0136 [
pdf]
An Algebraic Treatment of Congruences in Number Theory
In this article we will examine the behavior of certain free abelian subgroups of the multiplicative group of the positive rationals and their relationship with the group of units of integers modulo $n$.
[204]
vixra:2107.0135 [
pdf]
On the Lehmer's Totient Problem on Number Fields
Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.
[205]
vixra:2107.0121 [
pdf]
A Progress on the Binary Goldbach Conjecture
In this paper we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop series of steps to prove the binary Goldbach conjecture in full. We end the paper by proving the binary Goldbach conjecture for all even numbers exploiting the strategies outlined.
[206]
vixra:2107.0112 [
pdf]
A Topological Approach to the Twin Prime and De Polignac Conjectures
Abstract. We introduce a topology in the set of natural numbers via a subbase of open sets. With this topology, we obtain an irreducible (hyperconnected) space with no generic points. This fact allows proving that the cofinite intersections of subbasic open sets are always empty. That implies the validity of the Twin Prime Conjecture. On the other hand, the existence of strictly increasing chains of subbasic open sets shows that the Polignac Conjecture is false for an infinity of cases.
[207]
vixra:2107.0105 [
pdf]
An Identity Involving Tribonacci Numbers
In this paper, we present an identity involving Tribonacci Numbers. We will prove this identity by extending the number of variables of Candido's identity to three.
[208]
vixra:2107.0074 [
pdf]
On a Modular Property of Tetration
This paper generalizes problem 3 of the 2019 PROMYS exam, which asks to show that the last 10 digits (in base 10) of the n-th tetration of 3 are independent of ]n if n>10. The generalization shows that given any positive integers $a$ and b satisfying certain conditions, the last n digits (in base b) of the m-th tetration of a are independent of m if m>n. We use numerical patterns as a guide towards the solution and explore an additional numerical pattern which shows a relation between decimal expansions and multiplicative inverses of powers of 3 modulo powers of 10.
[209]
vixra:2107.0059 [
pdf]
On Prime Numbers in Linear Form
A lower bound is given for the number of primes in a special linear form less than N, under the assumption of the weakened Elliott-Halberstam conjecture.
[210]
vixra:2106.0076 [
pdf]
Riemann Hypothesis Proof Using Balazard, Saiasand Yor and Criterion
In this manuscript, we define a conformal map from the unit disc onto the semi plane. Later, we define a function f(z) = (s−1)ζ(s). We prove that f(z) belongs to the Hardy space,H1/3(D). We apply Jensen’s formula noting that the measure associated with the singularinterior factor of f is zero. Finally, we get∫∞−∞log|ζ(12+it)|14+t2dt=0.
[211]
vixra:2106.0009 [
pdf]
Proofs of Three Conjectures in Number Theory : Beal's Conjecture, Riemann Hypothesis and The $ABC$ Conjecture
This monograph presents the proofs of 3 important conjectures in the field of Number theory: - The Beal's conjecture. - The Riemann Hypothesis. - The $abc$ conjecture. We give in detail all the proofs.
[212]
vixra:2105.0180 [
pdf]
A New Inequality for the Riemann Hypothesis
There have been published many research results on the Riemann hypothesis. In this paper, we first find a new inequality for the Riemann hypothesis on the basis of well-known Robin theorem. Next, we introduce the error terms suitable to Mertens' formula and Chebyshev's function, and obtain their estimates. With such estimates and primorial numbers, we finally prove that the new inequality holds unconditionally.
[213]
vixra:2105.0118 [
pdf]
An Extension of the Erdos-Turan Additive Base Conjecture Via Generalized Circles of Partition
This paper is an extension program of the notion of circle of partition developed in our first paper [1]. As an application we prove the Erdos-Turan additive base conjecture.
[214]
vixra:2104.0172 [
pdf]
How to Find the Surplus Root (Prime Number) in Power Surplus of Prime Numbers
There are already various formulas for calculating power remainders and roots of remainders. Based on these, I have created a simple and quick way to calculate it. However, there is no theoretical proof.
[215]
vixra:2104.0080 [
pdf]
Another Look at "Faulhaber and Bernoulli"
Let "Faulhaber's formula" refer to an expression for the sum of powers of integers written with terms of n(n+1)/2. Initially, the author used Faulhaber's formula to explain why odd Bernoulli numbers are equal to zero. Next, Cereceda gave alternate proofs of that result and then proved the converse, if odd Bernoulli numbers are equal to zero then we can derive Faulhaber's formula. Here, the original author will give a new proof of the converse.
[216]
vixra:2104.0076 [
pdf]
On the Gap Sequence and Gilbreath's Conjecture
Motivated by Gilbreath's conjecture, we develop the notion of the gap sequence induced by any sequence of numbers. We introduce the notion of the path and associated circuits induced by an originator and study the conjecture via the notion of the trace and length of a path.
[217]
vixra:2104.0072 [
pdf]
Irrationality Proofs: From e to Zeta(n>=2)
We develop definitions and a theory for convergent series that have terms of the form $1/a_j$ where $a_j$ is an integer greater than one and the series convergence point is less than one. These series have terms with denominators that can be used as number bases. The series for $e-2$ and $z_n=\zeta(n)-1$ are of this type. Further, both series yield number bases that can represent all possible rational convergence points as single digits. As partials for these series are rational numbers, all partials can be given as single decimals using some $a_j$ as a base. In the case of $e-2$, the last term of a partial yields such a base and partials form systems of nesting inequalities yielding a proof of the irrationality of $e-2$. Using limits in an unusual way we are able to give a second proof for the irrationality of $e-2$. A third proof validates the second using Dedekind cuts. In the case of $z_n$, using the $z_2$ case we determine that such systems of nesting inequalities are not formed, but we discover partials require bases greater than the denominator of their last term. We prove this property for the general $z_n$ case and, using the unusual limit style proof mentioned, prove $z_n$ is irrational. We once again validate the proof using Dedekind cuts. Finally, we are able to give what we consider a satisfying proof showing why both $e-2$ and $z_n$ are irrational.
[218]
vixra:2104.0062 [
pdf]
Gentle Beukers's Proofs that Zeta(2,3) are Irrational
Although Beukers's proof that Zeta(2) and Zeta(3) are irrational are at the level of advanced calculus, they are condensed. This article slows down the development and adds examples of the techniques used. In so doing it is hoped that more people might enjoy these mathematical results. We focus on the easier of the two Zeta(2).
[219]
vixra:2104.0020 [
pdf]
Mathematical Modelling of COVID-19 and Solving Riemann Hypothesis, Polignac's and Twin Prime Conjectures Using Novel Fic-Fac Ratio With Manifestations of Chaos-Fractal Phenomena
COVID-19 originated from Wuhan, China in December 2019. Declared by the World Health Organization on March 11, 2020; COVID-19 pandemic has resulted in unprecedented negative global impacts on health and economy. International cooperation is required to combat this "Incompletely Predictable" pandemic. With manifestations of Chaos-Fractal phenomena, we mathematically model COVID-19 and solve [unconnected] open problems in Number theory using our versatile Fic-Fac Ratio. Computed as Information-based complexity, our innovative Information-complexity conservation constitutes a unique all-purpose analytic tool associated with Mathematics for Incompletely Predictable problems. These problems are literally "complex systems" containing well-defined Incompletely Predictable entities such as nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of using our unique Dimension (2x - N) system instead of this Sieve result in primary spin-offs from first key step consisting of providing proof for Riemann hypothesis (and explaining closely related two types of Gram points), and second key step consisting of providing proofs for Polignac's and Twin prime conjectures.
[220]
vixra:2103.0199 [
pdf]
Fermat's Last Theorem: A Simple Proof
This paper provides a simple proof of Fermat’s Last Theorem via elementary algebraic analysis of a level that would have been extant in Fermat’s day, the mid seventeenth century. The proof is effected by transforming Fermat's equation to an nth order polynomial, which is solved for 5 cases revealing a pattern that enables an extrapolation to the general case.
[221]
vixra:2103.0158 [
pdf]
The Collatz Conjecture and the Quantum Mechanical Harmonic Oscillator
By establishing a dictionary between the QM harmonic oscillator and the Collatz process, it reveals very important clues as to why the Collatz conjecture most likely is true. The dictionary requires expanding any integer $ n $ into a binary basis (bits) $ n = \sum a_{nl} 2^l $ ($l$ ranges from $ 0 $ to $ N - 1$) that allows to find the correspondence between every integer $ n $ and the state $ | \Psi_n \rangle $, obtained by a superposition of bit states $ | l \rangle $, and which are related to the energy eigenstates of the QM harmonic oscillator. In doing so, one can then construct the one-to-one correspondence between the Collatz iterations of numbers $ n \rightarrow { n \over 2 }$ ($n$ even); $ n \rightarrow 3 n + 1$ ($n$ odd) and the operators $ {\bf L}_{ { n \over 2} }; { \bf L}_{ 3 n + 1 } $, which map $ \Psi_n $ to $ \Psi_{ { n \over 2 } }$, or to $ \Psi_{ 3 n + 1 } $, respectively, and which are constructed explicitly in terms of the creation $ {\bf a}^\dagger$, annihilation $ {\bf a }$, and unit operator $ { \bf 1 } $ of the QM harmonic oscillator. A rigorous analysis reveals that the Collatz conjecture is most likely true, if the composition of a chain of $ {\bf L}_{ { n \over 2} }; { \bf L}_{ 3 n + 1 } $ operators (written as $ L_*$ in condensed notation) leads to the null-eigenfunction conditions $ ( {\bf L_* L_* \ldots L_* } - {\cal P } ) \Psi_n = 0 $, where $ {\cal P} $ is the operator that $projects$ any state $ \Psi_n $ into the ground state $ \Psi_1 \equiv | 0 \rangle $ representing the zero bit state $ | 0 \rangle$ (since $2^0 = 1$). In essence, one has a realization of the integer/state correspondence typical of QM such that the Collatz paths from $ n $ to $ 1$ are encoded in terms of quantum transitions among the states $ \Psi_n$, and leading effectively to an overall downward cascade to $ \Psi_1$. The QM oscillator approach explains naturally why the Collatz conjecture fails for negative integers because there are no states below the ground state.
[222]
vixra:2103.0150 [
pdf]
On the Infinitude of Sophie Germain Primes
In this paper we obtain the estimate \begin{align} \# \left \{p\leq x~|~2p+1,p\in \mathbb{P}\right \}\geq (1+o(1))\frac{\mathcal{D}}{(2+2\log 2)}\frac{x}{\log^2x}\nonumber \end{align}where $\mathbb{P}$ is the set of all prime numbers and $\mathcal{D}\geq 1$. This proves that there are infinitely many primes $p\in \mathbb{P}$ such that $2p+1\in \mathbb{P}$ is also prime.
[223]
vixra:2103.0136 [
pdf]
Numbers of Goldbach Conjecture Occurence in Every Even Numbers
This paper proposed proof of Goldbach Conjecture by using a function such that the numbers occurences of conjecture solution in any even numbers can be estimated. The function sketches after Eratoshenes Sieve under modulo term such that the function fulfilled prime sub-condition in closed intervals.
[224]
vixra:2102.0163 [
pdf]
A Generalization of Vajda's Identity for Fibonacci and Lucas Numbers
In this paper, we present two identities involving Fibonacci numbers and Lucas numbers. The first identity generalizes Vajda's identity, which in turn generalizes Catalan's identity, while the second identity is a corresponding result involving Lucas numbers. Binet's formulas for generating the nth term of Fibonacci numbers and Lucas numbers will be used in proving the identities.
[225]
vixra:2102.0161 [
pdf]
A Proof of Lemoine's Conjecture by Circles of Partition
In this paper we use a new method to study problems in additive number theory. We leverage this method to prove the Lemoine conjecture, a closely related problem to the binary Goldbach conjecture. In particular, we show by using the notion of circles of partition that for all odd numbers $n\geq 9$ holds \begin{align*} n=p+2q\mbox{ for not necessarily different primes }p,q. \end{align*}
[226]
vixra:2102.0064 [
pdf]
Affirmative Resolve of the Riemann Hypothesis
Riemann Hypothesis has been the unsolved conjecture for 170 years. This conjecture is the last one of conjectures without proof in "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse"(B.\\Riemann). The statement is the real part of the non-trivial zero points of the Riemann Zeta function is 1/2. Very famous and difficult this conjecture has not been solved by many mathematicians for many years. In this paper, I try to solve the proposition about the Mobius function equivalent to the Riemann Hypothesis. First, the non-trivial formula for Mobius function is proved in theorem 1 and theorem 2. In theorem 4, I get upper bound for the sum of the mobius functions (for meaning of R.H. See theorem 4).
[227]
vixra:2102.0044 [
pdf]
Solution to the Riemann Hypothesis from Geometric Analysis of Component Series Functions in the Functional Equation of Zeta
This paper presents a new approach towards the Riemann Hypothesis. On iterative expansion of integration term in functional equation of the Riemann zeta function we get sum of two series function. At the ‘nontrivial’ zeros of zeta function, value of the series is zero. Thus, Riemann hypothesis is false if that happens for an ‘s’ off the line <(s) = 1/2 ( the critical line). This series has two components f(s) and f(1 − s). For the hypothesis to be false one component is additive inverse of the other. From geometric analysis of spiral geometry representing the component series functions f(s) and f(1 − s) on complex plane we find by contradiction that they cannot be each other’s additive inverse for any s, off the critical line. Thus, proving truth of the hypothesis.
[228]
vixra:2101.0171 [
pdf]
The Binary Goldbach Conjecture and Circles of Partition
In this paper we use a new method to study problems in the additive number theory (see \cite{CoP}). With the notion of circle of partition as a set of points whose weights are natural numbers of a particular subset under an additive condition we are almost able to prove the binary Goldbach conjecture.
[229]
vixra:2101.0117 [
pdf]
Approximate Formula For zeta Function ζ(s) and L Function L(s) S=re
I created an approximate calculation formula for the zeta function and the L function. The range is 1 <x <2 and x> = 2. The L function has only ×> = 2. In both cases, the accuracy increases as the starting point moves away from 2.There is no mathematical proof.
[230]
vixra:2101.0079 [
pdf]
The Simple Condition of Fermat Wiles Theorem Mainly Led by Combinatorics
This paper gives the simple and necessary condition of Fermat Wiles Theorem with mainly providing one method to analyze natural numbers and the formula X^n + Y^n = Z^n logically and geometrically, which is positioned in combinatorial design theory. The condition is gcd(X, E)^n = X − E ∧ gcd(Y, E)^n = Y − E in ¬(n | XY ), or gcd(X, E)^n/n = X − E ∧ gcd(Y, E)^n = Y − E in n | X ∧ ¬(n | Y ). Provided that E denotes E = X + Y − Z, n is a prime number equal to or more than 2, and X, Y, Z are coprime numbers.
[231]
vixra:2012.0219 [
pdf]
Proving Zeta (n>=2) Is Irrational Using Decimal Sets
We prove that partial sums of Zeta(n)-1=zn are not given by any single decimal in a number base given by a denominator of their terms. These sets of single decimals we call decimal sets. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational, possible convergence points, elements of these decimal sets. The limit of the partials is zn and the limit of the exclusions leaves only irrational numbers. Thus zn is proven to be irrational.
[232]
vixra:2012.0212 [
pdf]
Using Decimal to Prove e is Irrational
Using for decimal bases the terms of e-2, we calculate partial sums and form open intervals for tails of partials. These intervals exclude all possible rational convergence points and thus show e-2 and hence e is irrational.
[233]
vixra:2012.0164 [
pdf]
Primorials and a Formula for Odd Abundant Numbers
We conjecture that a formula that represents a difference between two primorials of different parities generates only odd abundant numbers for its arguments greater than 3. Using PARI/GP, we verified this conjecture for the arguments up to $4*10^4$. We also discuss another formula that generates only odd abundant numbers in an arithmetic progression and explain its origin in the context of the distribution of odd abundant numbers in general.
[234]
vixra:2012.0163 [
pdf]
Smallest Numbers Whose Number of Divisors Is a Perfect Number
We present a formula for the smallest possible numbers whose number of divisors is the $n$-th perfect number. The formula, that produces an integer sequence $a(n)$, involves the $n$-th Mersenne prime that appears both in an exponent of a power of 2 and in the product of consecutive odd primes (the odd primorial). While smallest in some sense, these numbers are among largest one can run into through an exercise in elementary number theory.
[235]
vixra:2012.0119 [
pdf]
A Polynomial Pattern for Primes Based on Nested Residual Regressions
The pattern of the primes is one of the most fundamental mysteries of mathematics. This paper introduces a core polynomial model for primes based on nested residual regressions. Residual nestedness reveals increasing polynomial intertwining and shows scale invariance, or at least strong self-similarity up to at least p = 15,485,863. Accuracy of prediction decreases as the prediction range increases, conversely, the increase in the number of models helps refine predictions holistically.
[236]
vixra:2012.0106 [
pdf]
Is the ABC Conjecture True?
In this paper, we consider the abc conjecture. In the first part, we give the proof of the conjecture c < rad ^{1.63}(abc) that constitutes the key to resolve the abc conjecture. The proof of the abc conjecture is given in the second part of the paper, supposing that the abc conjecture is false, we arrive in a contradiction.
[237]
vixra:2012.0086 [
pdf]
Proof of the ABC Conjecture
In this short note, I prove the abc conjecture. You are free not to get enlightened about that fact. But please pay respect to new dispositions of the abc conjecture and research methods in this note.
[238]
vixra:2012.0039 [
pdf]
Proof of Riemann Hypothesis
This paper is a trial to prove Riemann hypothesis according to the following process. 1. We create the infinite number of infinite series from one equation that gives ζ(s) analytic continuation to Re(s) > 0 and 2 formulas (1/2 + a + bi, 1/2 − a − bi) which show zero point of ζ(s). 2. We find that a cannot have any value but zero from the above infinite number of infinite series. Therefore zero point of ζ(s) must be 1/2 ± bi.
[239]
vixra:2011.0212 [
pdf]
Assuming C Less Then Rad2 (Abc), a New Proof of the Abc Conjecture
In this paper, we consider the abc conjecture. Assuming that c<rad^2(abc) is true, we give a new proof of the abc conjecture, by proceeding with the contradiction of the definition of the abc conjecture, for \epsilon \geq 1, then for \epsilon \in ]0,1[.
[240]
vixra:2011.0199 [
pdf]
Acknowledgment of Non-linearity or How to Solve Several Conjectures
Several famous conjectures from Number Theory are studied. I derive a new equivalent formulation of Goldbach's strong conjecture and present an independent conjecture with some evidence for it.
[241]
vixra:2011.0198 [
pdf]
Exceptions from Robin's Inequality
In this short but rigorous research note I study Robin's Inequality. The number of possible violations of this inequality turns out to be finite. As the finiteness includes zero, I am able to convince you that there are no such violations.
[242]
vixra:2011.0138 [
pdf]
Modular Logarithm Unequal
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal logarithms in a corresponding module, and is applied to solving a kind of high degree diophantine equation.
[243]
vixra:2010.0222 [
pdf]
Assuming C Less Than Rad2(abc) Implies the ABC Conjecture Is True
In this paper about the $abc$ conjecture, assuming the condition $c<rad^2(abc)$ holds, and the constant $K(\epsilon)$ is a smooth function, having a derivative for $\epsilon \in ]0,1[$, then we give the proof of the $abc$ conjecture.
[244]
vixra:2010.0199 [
pdf]
Fibonacci-zeta Infinite Series Associated with the Polygamma Functions
We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.
[245]
vixra:2010.0189 [
pdf]
Isometric Admissibility for Bounded Subrings
Let H̃ be a right-irreducible, contra-characteristic, pairwise commutative manifold. In [23, 36, 15], the authors address the compactness of algebraic topoi under the additional assumption that − ζ̃(y) > ρ 00 0, . . . , BA( Ḡ) . We show that Φ > −1. The work in [35, 21] did not consider the associative case. In [17], the authors address the smoothness of real, Turing, sub-continuously D-Gödel random variables under the additional assumption that O > kT k.
[246]
vixra:2010.0146 [
pdf]
Super-Almost Countable Fields for a Functor
Let τ be an almost elliptic morphism. Recent interest in finitely right-free monoids has centered on constructing pseudo-analytically maximal groups. We show that every continuously negative system is degenerate, uncountable and invariant. In this setting, the ability to construct almost surely closed, almost surely hyper-empty homeomorphisms is essential. It would be interesting to apply the techniques of [11] to natural functors.
[247]
vixra:2010.0108 [
pdf]
Argumenting the Validity of Riemann Hypothesis
There are tens of self-proclaimed proofs for the Riemann Hypothesis and only 2 or 4 disproofs of it in arXiv. To this Status Quo I am adding my very short and clear results even without explicit mentioning prime numbers. One of my breakthroughs uses the peer-reviewed achievement of Dr.~Sol\'e and Dr.~Zhu, published just 4 years ago in a serious mathematical journal INTEGERS.
[248]
vixra:2010.0105 [
pdf]
Fibonacci Series from Power Series
We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.
[249]
vixra:2010.0035 [
pdf]
On a Modular Property of Odd Numbers Under Tetration
The aim of this paper is to generalize problem 3 of the 2019 PROMYS exam, which asks to show that the last 10 digits (in base 10) of t_n are same for all n >= 10, where t_0 = 3 and t_(k+1) = 3^(t_k). The generalization shows that given any positive odd integer p, t_m is congruent to t_n modulo [(p^2)+1]^n for all m >= n >= 1, where t_0 = p and t_(k+1) = p^(t_k)
[250]
vixra:2009.0195 [
pdf]
The Super-Generalised Fermat Equation Pa^x + Qb^y=rc^z and Five Related Proofs
In this paper, we consider five proofs related to the super-generalised Fermat equation, Pa^x + Qb^y=Rc^z. All proofs depend on a new identity for a^x + b^y which can be expressed as a binomial sum to an indeterminate power, z. We begin with the Generalised Fermat Conjecture, for the case P,Q,R=1, also known as the Tijdeman-Zagier Conjecture and Beal Conjecture. We then show how the method applies to its famous corollary Fermat's Last Theorem, where x,y,z=n. We then return to the title equation, considered by Henri Darmon and Andrew Granville and extend the proof for the case P,Q,R>1 and x,y,z>2. Finally, we use the results to prove Catalan's Conjecture, and from this a weak proof that under certain conditions only one solution exists for equations of the form a^4-c^2=b^y.
[251]
vixra:2009.0166 [
pdf]
The Circle Embedding Method and Applications
In this paper we introduce and develop the circle embedding method. This method hinges essentially on a Combinatorial structure which we choose to call circles of partition. We provide applications in the context of problems relating to deciding on the feasibility of partitioning numbers into certain class of integers. In particular, our method allows us to partition any sufficiently large number $n\in\mathbb{N}$ into any set $\mathbb{H}$ with natural density greater than $\frac{1}{2}$. This possibility could herald an unprecedented progress on class of problems of similar flavour. The paper finishes by giving a partial proof of the binary Goldbach conjecture.
[252]
vixra:2009.0068 [
pdf]
New Formula for Prime Counting Function
This paper presents two functions for prime counting function and its inverse function (the function that returns nth prime number as output) with high accuracy and best approximation, which, due to their significant features, are distinguished from other similar functions presented thus far. the presented function for prime counting function is denoted by πm(x) and presented function for nth prime is denoted by Pm(x) in this article.
[253]
vixra:2008.0182 [
pdf]
Une Note Sur La Conjecture ABC
It is a paper of Gerhard Frey published in 2012. It is an introduction about the $abc$ conjecture and its subtleties and consequences for the theory of numbers. It is a scientific version of the original paper.
[254]
vixra:2007.0196 [
pdf]
There Exist Infinitely Many Couples of Primes (P,p+2n) ,with 2n >2 is a Fixed Distance Between P and P+2n
We will prove the next results : 1. there exist infinite twin primes . 2. there exist infinite cousin primes . 3. The cousin primes are equivalent to twin primes in infinity.
[255]
vixra:2007.0128 [
pdf]
A Remark on the Strong Goldbach Conjecture
Under the assumption that $\sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)>0$, we show that for all even number $N>6$ \begin{align} \sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)=(1+o(1))K\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)\nonumber \end{align}for some constant $K>0$, and where $\Upsilon$ and $\Lambda_{0}$ denotes the master and the truncated Von mangoldt function, respectively. Using this estimate, we relate the Goldbach problem to the problem of showing that for all $N>6$ $(N\neq 2p)$, If $\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$, then $\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$ for each prime $p|N$.
[256]
vixra:2007.0116 [
pdf]
Riemann Hypothesis: New Criterion, Evidence, and One-Page Proof
There are tens of self-proclaimed proofs for the Riemann Hypothesis and only 2 or 4 disproofs of it in arXiv. I am adding to the Status Quo my very short and clear results even without explicit mentioning of the prime numbers. One of my breakthroughs uses the peer-reviewed achievement of Dr.Sole and Dr.Zhu, published just 4 years ago in a serious mathematical journal INTEGERS.
[257]
vixra:2007.0115 [
pdf]
Proofs for Goldbach's, Twin Prime, and Polignac's Conjectures
I derive a new equivalent formulation of Goldbach's strong conjecture and present several proofs of Goldbach's strong conjecture and other conjectures. You are free not to get enlightened about that facts. But please pay respect to new dispositions of the conjectures and research methods in this note.
[258]
vixra:2007.0102 [
pdf]
Bounds on the Range(s) of Prime Divisors of a Class of Baillie-PSW Pseudo-Primes
In the literature [1], Carmichael Numbers that satisfy additional constraints $(p+1) \mydivides (N+1)$ for every prime divisor $p \mydivides N$ are referred to as ``Williams' Numbers''\footnote{more precisely, ``1-Williams Numbers''~; however~; the distinctions between different types of Willliams' numbers are not relevant in this document and therefore, we refer to 1-Williams Numbers~ simply as Williams' numbers.}. % In the renowned Pomerance-recipe~\cite{pomerance1984there} to search for Baillie-PSW pseudoprimes; there are heuristic arguments suggesting that the number of Williams' Numbers could be large (or even unlimited). Moreover, it is shown~\cite{pomerance1984there} that if a Williams' number is encountered during a search in accordance with all of the conditions in that recipe~\cite{pomerance1984there}~; then it must also be a Baillie-PSW pseudoprime. We derive new analytic bounds on the prime-divisors of a Williams' Number.\\ Application of the bounds to Grantham's set of 2030 primes~(see ~\cite{grantham-620-list}) drastically reduces the search space from the impossible size $\approx 2^{(2030)}$ to less than a quarter billion cases (160,681,183 cases to be exact, please see the appendix for details). We tested every single case in the reduced search space with maple code. The result showed that there is \underline{NO Williams' number (and therefore NO Baillie-PSW pseudo-prime which is also a Williams' number)} in the entire space of subsets of the Grantham-set. The results thus demonstrate that Williams' numbers either do not exist or are extremely rare. We believe the former; i.e., that No such composite (i.e., a Williams' Number of this type) exists.
[259]
vixra:2007.0090 [
pdf]
Proof of Goldbach Conjecture
In the letter sent by Goldbach to Euler in 1742 (Christian, 1742) he stated that its seems that every odd number greater than 2 can be expressed as the sum of three primes. As reformulated by Euler, an equivalent form of this conjecture called the strong or binary Goldbach conjecture states that all positive even integers greater or equal to 4 can be expressed as the sum of two primes which are sometimes called a Goldbach partition. Jorg (2000) and Matti (1993) have verified it up to 4.1014. Chen (1973) has shown that all large enough even numbers are the sum of a prime and the product of at most two primes... The majority of mathematicians believe that Goldbach's conjecture is true, especially on statistical considerations ,on the subject we give the proof of Goldbach's strong conjecture whose veracity is based on a clear and simple approach.
[260]
vixra:2007.0042 [
pdf]
Some Relations Among Pythagorean Triples
Some relations among Pythagorean triples are established. The main tool is a fundamental characterization of the Pythagorean triples through a cathetus that allows to determine the relationships between two Pythagorean triples with an assigned cathetus a and b and the Pythagorean triple with cathetus a · b.
[261]
vixra:2006.0254 [
pdf]
Ulam Numbers Have Zero Density
In this paper we show that the natural density $mathcal{D}[(U_m)]$ of Ulam numbers $(U_m)$ satisfies $mathcal{D}[(U_m)]=0$. That is, we show that for $(U_m)subset [1,k]$ then begin{align}lim limits_{klongrightarrow infty}frac{left |(U_m)cap [1,k]ight |}{k}=0.onumberend{align}
[262]
vixra:2006.0184 [
pdf]
Sums of Powers of Fibonacci and Lucas Numbers and their Related Integer Sequences
In this paper we will look at sums of odd powers of Fibonacci and Lucas numbers of even indices. Our motivation will be conjectures, now theorems, which go back to Melham. Using the simple approach of telescoping sums we will be able to give new proofs of those results. Along the way we will establish inverse relationships for such sums and discover new integer sequences.
[263]
vixra:2006.0053 [
pdf]
On The Infinity of Twin Primes and other K-tuples
The paper uses the structure and math of Prime Generators to show there are an infinity of twin primes, proving the Twin Prime Conjecture, as well as establishing the infinity of other k-tuples of primes.
[264]
vixra:2006.0046 [
pdf]
An Improved Lower Bound of Heilbronn's Triangle Problem
Using the method of compression we improve on the current lower bound of Heilbronn's triangle problem. In particular, by letting $Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a unit disc. Then we have the lower boundbegin{align}Delta(s)gg frac{log s}{ssqrt{s}}.onumberend{align}
[265]
vixra:2005.0282 [
pdf]
A New Criterion for Riemann Hypothesis or a True Proof?
There are tens of self-proclaimed proofs for Riemann Hypothesis and only 2 or 4 disproofs of it in arXiv. I am adding to the Status Quo my very short and clear evidence which uses the peer-reviewed achievement of Dr.Sole and Dr.Zhu, which they published just 4 years ago in a serious mathematical journal INTEGERS.
[266]
vixra:2005.0269 [
pdf]
Maximality Methods in Commutative Set Theory
Let f = w be arbitrary. Every student is aware that Kolmogorov’s criterion applies. We show that S ≤ |ρR,C |. J. Sasaki [34] improved upon the results of R. Thomas by deriving subsets. The goal of the present paper is to compute irreducible, generic random variables
[267]
vixra:2005.0267 [
pdf]
Improved Estimate for the Prime Counting Function
Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}\pi(x)=\Theta(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber \end{align}where \begin{align}\Theta(x)=\frac{\theta(x)}{\log x}+\frac{x}{2\log x}-\frac{1}{4}-\frac{\log 2}{\log x}\sum \limits_{\substack{n\leq x\\\Omega(n)=k\\k\geq 2\\2\not| n}} \frac{\log (\frac{x}{n})}{\log 2}.\nonumber \end{align}This is an improvement to the estimate \begin{align}\pi(x)=\frac{\theta(x)}{\log x}+O\bigg(\frac{x}{\log^2 x}\bigg)\nonumber \end{align}found in the literature.
[268]
vixra:2005.0258 [
pdf]
One page Proof of Riemann Hypothesis
There are tenths of proofs for Riemann Hypothesis and 3 or 5 disproofs of it in arXiv. I am adding to the Status Quo my proof, which uses the achievement of Dr. Zhu.
[269]
vixra:2005.0237 [
pdf]
Proof of the Beal Conjecture and Fermat Catalan Conjecture (Summary)
This article inclucles the theorems anh the lemmas, using them to prove the Beal conjecture anh the Fermat- Catalan conjecture, through which we learn more about rational and irrational numbers. I think the method of proof will be useful for solving other Math- problems and they need more research.
[270]
vixra:2005.0236 [
pdf]
On the Pointwise Periodicity of Multiplicative and Additive Functions
We study the problem of estimating the number of points of coincidences of an idealized gap on the set of integers under a given multiplicative function $g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function $f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds depending on the length of the period, by varying the worst growth rates of the ratios of their consecutive values
[271]
vixra:2005.0208 [
pdf]
An Unintentional Repetition of the Ramanujan Formula for $\pi$, and Some Independent Mathematical Mnemonic Tools for Calculating with Exponentiation and with Dates
The paper is consisted of contentually unrelated sections, where each section could be one short paper. Section 1 deals with the process of an unintentional repetition of one of the Ramanujan formulas, and the author did not know it before. The process of the unintentional repetition of the Ramanujan formula is interesting for estimating, for instance, the physical background for guessing of dimensionless physical constants. Section 2 shows an approximation which helps at memorizing the square roots of integers up to 10. Section 3 shows the specialities of the squares of integers at the last digits. Section 4 shows how the last two digits are repeated for 2 to the sequential integer powers, and how to find out this. Section 5 contains some mathematical peculiarities at calculating dates. All sections contain mnemonic methods to help us memorize and calculate. These sections belong to pedagogical and recreational mathematics, maybe even something more is here.
[272]
vixra:2005.0201 [
pdf]
The abc Conjecture is False: The End of The Mystery
In this note, I give the proof that the abc conjecture is false because, in the case c>rad(abc), for 0<\epsilon<1, presenting a counterexample that implies a contradiction for c very large.
[273]
vixra:2005.0130 [
pdf]
On the Existence of Prime Numbers in Constant Gaps
This paper studies the existence of prime numbers on constant gaps, establishing a lower bound for the number of consecutive constant gaps for which the existence of some prime number contained in them is necessary.
[274]
vixra:2005.0076 [
pdf]
The Fermat Classes and the Proof of Beal Conjecture
If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles , The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by using the Fermat class concept.
[275]
vixra:2005.0059 [
pdf]
The Special Functions and the Proof of the Riemann’s Hypothesis
By studying the $ \circledS $ function whose integer zeros are the prime numbers, and being inspired by the article [2], I give a new proof of the Riemann hypothesis.
[276]
vixra:2005.0004 [
pdf]
Assuming C<rad^2(abc) :A Proof of the Abc Conjecture
In this paper, assuming the conjecture $c<rad^2(abc)$ true, I give, using elementary calculus, the proof of the $abc$ conjecture proposing the constant $K(\ep)$. Some numerical examples are given.
[277]
vixra:2004.0690 [
pdf]
A^x + B^y = C^z Part 2: Another Version of my Theorem and Infinite Ascent
We give another version of my theorem that was submitted in the previous article, which is the theorem leading to the proof of the Beal's conjecture and the Fermat - Catalan conjecture. We also give a view to prove whether the equation has infinitely many solutions in integer or not related to parametric solution and infinite ascent.
[278]
vixra:2004.0294 [
pdf]
Numbers Are 3 Dimensional
Riemann hypothesis stands proved in three different ways of three different level of complexity.To prove Riemann hypothesis from the functional equation concept of Delta function and periodic harmonic conjugate of both Gamma and Delta functions are introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing zeta function to an extended domain, poles and zeros of zeta values are redefined. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be three dimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function and periodic harmonic conjugate of both Gamma and Delta functions.
[279]
vixra:2004.0281 [
pdf]
Quantom Test of Prime Numbers
we give a new method to for testing whether any positive integer is prime or not using real experiment by throwing neutron into a Plutonium.
[280]
vixra:2004.0246 [
pdf]
On the Distribution of Addition Chains
In this paper we study the theory of addition chains producing any given number $n\geq 3$. With the goal of estimating the partial sums of an additive chain, we introduce the notion of the determiners and the regulators of an addition chain and prove the following identities\begin{align}\sum \limits_{j=2}^{\delta(n)+1}s_j=2(n-1)+(\delta(n)-1)+\kappa(a_{\delta(n)})-\varrho(r_{\delta(n)+1})+\int \limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}\varrho(r_j)dt\nonumber \end{align}where \begin{align}2,s_3=\kappa(a_3)+\varrho(r_3),\ldots,s_{k-1}=\kappa(a_{k-1})+\varrho(r_{k-1}),s_{k}=\kappa(a_{k})+\varrho(r_{k})=n\nonumber \end{align}are the associated generators of the chain $1,2,\ldots,s_{k-1},s_{k}=n$ of length $\delta(n)$. Also we obtain the identity\begin{align}\sum \limits_{j=2}^{\delta(n)+1}\kappa(a_j)=(n-1)+(\delta(n)-1)+\kappa(a_{\delta(n)})-\varrho(r_{\delta(n)+1})+\int \limits_{2}^{\delta(n)-1}\sum \limits_{2\leq j\leq t}\varrho(r_j)dt.\nonumber \end{align}
[281]
vixra:2004.0169 [
pdf]
The $abc$ Conjecture: the Proof of $c<rad^2(abc)$
In this note, I present a very elementary proof of the conjecture $c<rad^2(abc)$ that constitutes the key to resolve the $abc$ conjecture. The method concerns the comparison of the number of primes of $c$ and $rad^2(abc)$ for large $a,b,c$ using the prime counting function $\pi(x)$ giving the number of primes $\leq x$. Some numerical examples are given.
[282]
vixra:2003.0494 [
pdf]
The Area Method and Applications
In this paper we develop a general method for estimating correlations of the forms \begin{align}\sum \limits_{n\leq x}G(n)G(x-n),\nonumber \end{align}and \begin{align}\sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. To distinguish between the two types of correlations, we call the first \textbf{type} $2$ correlation and the second \textbf{type} $1$ correlation. As an application we estimate the lower bound for the \textbf{type} $2$ correlation of the master function given by \begin{align}\sum \limits_{n\leq x}\Upsilon(n)\Upsilon(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x,\nonumber \end{align}provided $\Upsilon(n)\Upsilon(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture by showing that \begin{align}\sum \limits_{n\leq x}\Lambda(n)\Lambda(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align}for some $\mathcal{C}:=\mathcal{C}(2)>0$.
[283]
vixra:2003.0090 [
pdf]
Optimal Binary Number System When Numbers Are Energy?
In this short note, we will quickly look at optimal binary number systems used in communication (or transactions) under the assumption that one must use energy to give away (send) numbers. We show that the current binary system is not the optimal binary number system as it can be arbitraged. We also show that there exist other optimal binary number systems in such a scenario. Naturally, one has to ask, ``Optimal for whom? -- For the one sending the number out, or for the one receiving the number?'' Alternatively, we can have a binary number system that, on average, is neutral for both sender and receiver. Numbers are typically only considered to have symbolic value, but if the money units were so small that they came in the smallest possibly energy units, then we could be forced to switch to a number system where the physical value of each number was equal to its symbolic value. That is to say, the physical value of three must be higher than the physical value of two, for example. Numbers are always physical because storing or sending a number from a computer requires bits, and bits of information require energy.
[284]
vixra:2003.0066 [
pdf]
An Elementary Proof of Goldbach's Conjecture
Goldbach's conjecture is proven using the Chinese Remainder Theorem. It is shown that an even number 2N greater than four cannot exist if it is congruent to every prime p less than N (mod a different prime number).
[285]
vixra:2003.0050 [
pdf]
A Generator for Sums of Powers of Recursive Integer Sequences
In this paper we will prove a relationship for sums of powers of recursive integer sequences. Also, we will give a possible path to discovery. As corollaries of the main result we will derive relationships for familiar integer sequences like the Fibonacci, Lucas, and Pell numbers. Last, we will discuss some applications and point to further work.
[286]
vixra:2003.0008 [
pdf]
On the Erdh{o}s Distance Problem
In this paper, using the method of compression, we recover the lower bound for the ErdH{o}s unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $mathbb{R}^k$ for all $kgeq 2$, we have begin{align}# bigg{||vec{x_j}-vec{x_t}||:~||vec{x_j}-vec{x_t}||=1,~1leq t,j leq n,~vec{x_j},~vec{x}_t in mathbb{R}^kbigg}gg_k frac{sqrt{k}}{2}n^{1+o(1)}.onumberend{align}We also show thatbegin{align}# bigg{d_j:d_j=||vec{x_s}-vec{y_t}||,~d_jeq d_i,~1leq s,tleq nbigg}gg_k frac{sqrt{k}}{2}n^{frac{2}{k}-o(1)}.onumber end{align}These lower bounds generalizes the lower bounds of the ErdH{o}s unit distance and the distinct distance problem to higher dimensions.
[287]
vixra:2002.0466 [
pdf]
New Clues on Arbitrary-Precision Calculation of the Riemann Zeta Function On The Critical Line
The Riemann Hypothesis, is considered by many mathematicians to be the most important unsolved problem, consist in the assertion that all of zeta's nontrivial zeros line up at the so called critical line, $\zeta(1/2+it)$. This paper presents an algorithm, based on a closed-form system of equations, that computes directly at $n^{th}$ decimal digit each non-trivial zeros of the Riemann Zeta Function.
[288]
vixra:2002.0396 [
pdf]
Riemann Hypothesis and the Zeroes of Riemann Zeta Function
The proof involves analytic continuation of Riemann Zeta function. Further we work on the Hadamard product representation of Riemann Xi function to prove the Riemann Hypothesis
[289]
vixra:2002.0256 [
pdf]
Assuming C<R.exp(\frac{3\sqrt[3]{2}}{2}Log^{2/3}R) a New Conjecture Implies the Abc Conjecture True
In this paper about the abc conjecture, we propose a new conjecture about an upper bound for c as c<R.exp(\frac{3\sqrt[3]{2}}{2}Log^{2/3}R). Assuming the last condition holds, we give the proof of the abc conjecture by proposing the expression of the constant K(\epsilon), then we approve that \forall \epsilon>0, for a,b,c positive integers relatively prime with c=a+b, we have c< K(\epsilon).rad^{1+\epsilon}(abc). Some numerical examples are given.
[290]
vixra:2001.0690 [
pdf]
Naturally Numbers Are Three Plus One Dimensional Final Version 31.01.2020
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
[291]
vixra:2001.0655 [
pdf]
On the Number of Monic Admissible Polynomials in the Ring $\mathbb{z}[x]$
In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular, letting $\mathcal{N}(H)$ denotes the number of monic admissible polynomials of degree $n\geq 3$ with coefficients satisfying the inequality $0\leq a_i\leq H$, we show that \begin{align}\frac{H^{n-1}}{(n-1)!}+O(H^{n-2})\leq \mathcal{N}(H) \leq \frac{n^{n-1}H^{n-1}}{(n-1)!}+O(H^{n-2}).\nonumber \end{align} Also letting $\mathcal{A}(H)$ denotes the number of monic irreducible admissible polynomials, with coefficients satisfying the same condition , we show that \begin{align}\mathcal{A}(H)\geq \frac{H^{n-1}}{(n-1)!}+O\bigg( H^{n-4/3}(\log H)^{2/3}\bigg).\nonumber \end{align}
[292]
vixra:2001.0654 [
pdf]
The Prime Index Function
In this paper we introduce the prime index function \begin{align}\iota(n)=(-1)^{\pi(n)},\nonumber \end{align} where $\pi(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by\begin{align}\xi(x):=\sum \limits_{n\leq x}\iota(n).\nonumber \end{align}
[293]
vixra:2001.0653 [
pdf]
Complete Sets
In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem of counting the number of complete subsets of any given set. That is, given any interval of integers $\mathcal{H}:=[1,N]$ and letting $\mathcal{C}(N)$ denotes the complete set counting function, we establish the lower bound $\mathcal{C}(N)\gg N\log N$.
[294]
vixra:2001.0474 [
pdf]
The Prime Pairs Are Equidistributed Among the Coset Lattice Congruence Classes
In this paper we show that for some constant $c>0$ and for any $A>0$ there exist some $x(A)>0$ such that, If $q\leq (\log x)^{A}$ then we have \begin{align}\Psi_z(x;\mathcal{N}_q(a,b),q)&=\frac{\Theta (z)}{2\phi(q)}x+O\bigg(\frac{x}{e^{c\sqrt{\log x}}}\bigg)\nonumber \end{align}for $x\geq x(A)$ for some $\Theta(z)>0$. In particular for $q\leq (\log x)^{A}$ for any $A>0$\begin{align}\Psi_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{2\phi(q)}\nonumber \end{align}for some constant $\mathcal{D}(z)>0$ and where $\phi(q)=\# \{(a,b):(p_i,p_{i+z})\in \mathcal{N}_q(a,b)\}$.
[295]
vixra:2001.0472 [
pdf]
The Compression Method and Applications
In this paper we introduce and develop the method of compression of points in space. We introduce the notion of the mass, the rank, the entropy, the cover and the energy of compression. We leverage this method to prove some class of inequalities related to Diophantine equations. In particular, we show that for each $L<n-1$ and for each $K>n-1$, there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align}\frac{1}{K^{n}}\ll \prod \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{\log (\frac{n}{L})}{nL^{n-1}}\nonumber \end{align}and that for each $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ and some $s\geq 2$ such that \begin{align}\sum \limits_{j=1}^{n}\frac{1}{x_j^s}\gg s\frac{n}{L^{s-1}}.\nonumber \end{align}
[296]
vixra:2001.0204 [
pdf]
The Theory of the Collatz Process
In this paper we introduce and develop the theory of the Collatz process. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie germain primes. We also provide several formulation of the Collatz conjecture in this language.
[297]
vixra:2001.0152 [
pdf]
Assuming C<rad^2(abc), the Abc Conjecture is True
In this paper, assuming that c<rad^2(abc) is true, we give a proof of the abc conjecture by proposing the expression of the constant K(\epsilon), then we approve that \forall \epsilon>0, for a,b,c positive integers relatively prime with c=a+b, we have c<K(\epsilon).rad(abc)^{1+\epsilon}. Some numerical examples are given.
[298]
vixra:2001.0151 [
pdf]
Naturally Numbers Are Three Plus One Dimensional Final
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
[299]
vixra:2001.0097 [
pdf]
Definitive Tentative of a Proof of the \textit{abc} Conjecture
In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof that $c<3rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=\frac{3}{e}.e^{ \left(\frac{1}{\epsilon^2} \right)}$ for $0<\epsilon <1$ and $K(\epsilon)=3$ for $\epsilon \geq 1$. Some numerical examples are presented.
[300]
vixra:1912.0540 [
pdf]
A Remark on the Erd\'{o}s-Straus Conjecture
In this paper we discuss the Erd\'{o}s-Straus conjecture. Using a very simple method we show that for each $L\in \mathbb{N}$ with $L>n-1$ there exist some $(x_1,x_2,\ldots,x_n)\in \mathbb{N}^n$ with $x_i\neq x_j$ for all $1\leq i<j\leq n$ such that \begin{align}\frac{n}{L}\ll \sum \limits_{j=1}^{n}\frac{1}{x_j}\ll \frac{n}{L}\nonumber \end{align}In particular, for each $L\geq 3$ there exist some $(x_1,x_2,x_3)\in \mathbb{N}^3$ with $x_1\neq x_2$, $x_2\neq x_3$ and $x_3\neq x_1$ such that \begin{align}c_1\frac{3}{L}\leq \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}\leq c_2\frac{3}{L}\nonumber \end{align}for some $c_1,c_2>1$.
[301]
vixra:1912.0538 [
pdf]
The Little $\ell$ Function
In this short note we introduce a function which iteratively behaves in a similar fashion compared to the factorial function. However the growth rate of this function is not as dramatic and sudden as the factorial function. We also propose an approximation for this function for any given input, which holds for sufficiently large values of n.
[302]
vixra:1912.0531 [
pdf]
The Connection Between X^2+1 and Balancing Numbers
Balancing numbers as introduced by Behera and Panda [1] can be shown to be connected to the formula x^2+1=N in a very simple way. The goal of this paper is to show that if a balancing number exists for the balancing equation 1+ 2+ ... + (y-1) = (y+1)+(y+2)+...+(y+m), then there is a corresponding(2y)^2+1=N, where N is composite. We will also show how this can be used to factor N.
[303]
vixra:1912.0528 [
pdf]
A Proof of the Twin Prime Conjecture
In this paper we prove the twin prime conjecture by showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align}where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed and $\mathbb{P}$ is the set of all prime numbers. In particular it follows that \begin{align} \sum \limits_{p,p+2\in \mathbb{P}}1=\infty\nonumber \end{align}by taking $x\longrightarrow \infty$ on both sides of the inequality. We start by developing a general method for estimating correlations of the form \begin{align} \sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align}for a fixed $1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$.
[304]
vixra:1912.0526 [
pdf]
Naturally Numbers Are Three Plus One Dimensional
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined. Hodge conjecture, BSD conjecture are also proved using zeta values. Other prime conjectures like Goldbach conjecture, Twin prime conjecture etc.. are also proved in the light of new understanding of primes. Numbers are proved to be multidimensional as worked out by Hamilton. Logarithm of negative and complex numbers are redefined using extended number system. Factorial of negative and complex numbers are redefined using values of Delta function.
[305]
vixra:1912.0494 [
pdf]
Twin Prime Conjecture(newer Version)
I proved the Twin Prime Conjecture. The probability twin prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in. I investigated up to 5$\times10^{12}$.\\ When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\ However, it is not 0. Because, primes continue to be produced. Therefore, Twin Primes continue to be produced.\\ If the Twin Primes is finite, the primes is finite.\\ This is because slightly lower than 4/3 times the square of the probability of primes is the probability of Twin Primes.\\ This is contradiction. Because there are an infinite of primes.\\ \ \\ $[Probability\ of\ the\ Existence\ of\ primes]^2\times4/3$=\\ (Probability\ of\ the\ Existence\ of\ Twin\ Primes)\\ When the number becomes extreme, the generation of prime numbers becomes extremely small. However, it is not 0.\\ Very few, but prime numbers are generated.\\ Therefore, even if the number reaches the limit, twin prime numbers are also generated.\\ That is, Twin Primes exist forever.\\
[306]
vixra:1912.0225 [
pdf]
Numbers Are Three Dimensional, as Nature
Riemann hypothesis stands proved in three different ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Other two proofs are derived using Eulers formula and elementary algebra. Analytically continuing gamma and zeta function to an extended domain, poles and zeros of zeta values are redefined.
[307]
vixra:1912.0207 [
pdf]
Consideration of Twin Prime Conjecture\\ Average Difference is 2.296
I considered the Twin Prime Conjecture. The probability twin prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in.\\ When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\ The places where prime numbers come out are filled with multiples of primes one after another, and eventually disappear almost.\\ Primes can only occur very rarely when the numbers are huge.\\ This is natural from the following equation.\\ \begin{equation} \pi(x)\sim\frac{x}{\log{x}}\ \ \ (x\to\infty) \end{equation}\\ $[Probability\ of\ the\ Existence\ of\ primes]^2\times4/3\sim$ (Probability\ of\ the\ Existence\ of\ Twin\ Primes)\\ When the number becomes extreme, the generation of primes becomes extremely small. However, it is not 0.\\ Very few, but primes are generated.\\ If the twin primes appears as two primes completely independently, Twin Prime Problem is denied.\\ However, if twin primes appear in combination and appear like primes, twin primes consist forever and Twin Prime Problem is correct.\\
[308]
vixra:1912.0205 [
pdf]
Almost no Primes in the Infinite World
There are almost no primes in the infinite world. This is because the place where the primes appears is occupied by multiple of the primes. If you think about a hexagon, you can see it right away.
[309]
vixra:1912.0174 [
pdf]
Special Value of Riemann Zeta Function and L Function, Approximate Calculation Formula of ζ(N), L(N)
I made an approximate formula. In the formula, when N is small, the accuracy is very bad, but as N increases, the accuracy also improves.
[310]
vixra:1912.0157 [
pdf]
A proof of Twin Prime Conjecture
I proved the Twin Prime Conjecture. The probability that (6n -1) is a prime and (6n+1) is also a prime approximately is slightly lower than 4/3 times the square of the probability that a prime will appear in. I investigated up to 5$\times10^{12}$.\\ All Twin Primes are produced in hexagonal circulation. It does not change in a huge number (forever huge number).\\ The production of Twin Primes equal the existence of Twin Primes.\\ When the number grows to the limit, the primes to be produced rarely, but since Twin Primes are slightly lower than 4/3 times the square of the distribution of primes, the frequency of production of Twin Primes is very equal to 0.\\ However, it is not 0. Because, primes continue to be produced. Therefore, Twin Primes continue to be produced.\\ If the Twin Primes is finite, the primes is finite.\\ This is because slightly lower than 4/3 times the square of the probability of primes is the probability of Twin Primes. This is contradiction. Because there are an infinite of primes.\\
[311]
vixra:1912.0151 [
pdf]
A Proof of Twin Prime Conjecture by 30 Intervals Etc.
If (p,p+2) are twin primes, (p+30, p+2+30) or (p+60, p+2+60) or (p+90, p+2+90) or (p+120, p+2+120) or (p+150, p+2+150) ) or (p+180, p+2+180) or (p+210, p+2+210) or (p+240, p+2+240)……. is to be a twin primes.\\ There are three type of twin primes, last numbers are (1, 3)..(7, 9)..(9, 1).\\ They are lined up at intervals such as 30 or 60 or 90 or 120 or 150 or 180 or 210 or 240 or 270 or 300 etc.\ That is, it is a multiple of 30. \\ Repeat this.\\ And the knowledge about prime numbers is also taken into account.\\ That is, Twin Primes exist forever.\\
[312]
vixra:1912.0119 [
pdf]
Simple Prime Number Determination Method for Natural Numbers Including Carmichael Numbers
Explanation of effective prime number judgment method even for Carmichael number. This method of judgment does not give a 100% correct answer. Care must be taken especially for (n=p^k (P=Prime)) with primitive roots.
[313]
vixra:1911.0477 [
pdf]
Evidence, that X^2+y^3=1 and Others Have no Solution in Q>0
Due to the Incompleteness Theorems of Gödel one can say, that some true conjectures do not have valid proofs. One could think it also about my conjectures below, but I was lucky to find evidence for them.
[314]
vixra:1911.0316 [
pdf]
The Prime Counting Function and the Sum of Prime Numbers
In this paper it is proved that the sum of consecutive prime numbers up to the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, they are found some solutions such that both series are equal. Finally, they are listed the prime numbers at which both series are equal, and exposed some conjectures regarding this type of prime numbers.
[315]
vixra:1911.0287 [
pdf]
Recurring Pairs of Consecutive Entries in the Number-of-Divisors Function
The Number-of-Divisors Function tau(n) is the number of divisors of a positive integer n, including 1 and n itself. Searching for pairs of the format (tau(n), tau(n+1)), some pairs appear (very) often, some never and some --- like (1,2), (4,9), or (10,3) --- exactly once. The manuscript provides proofs for 46 pairs to appear exactly once and lists 12 pairs that conjecturally appear only once. It documents a snapshot of a community effort to verify sequence A161460 of the Online Encyclopedia of Integer Sequences that started ten years ago.
[316]
vixra:1911.0180 [
pdf]
Prime Sextuplet Conjecture
Prime Sextuplet and Twin Primes have exactly the same dynamics. All Prime Sextuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number). In the hexagon, Prime Sextuplet are generated only at (6n -1)(6n+5). [n is a positive integer] When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Sextuplet are 48/3 times of the sixth power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0. However, it is not 0. Therefore, Prime Sextuplet continue to be generated. If Prime Sextuplet is finite, the Primes is finite. The probability of Prime Sextuplet 48/3 times of the sixth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes. That is, Prime Sextuplet exist forever.
[317]
vixra:1911.0179 [
pdf]
Prime Quintuplet Conjecture
Prime Quintuplet and Twin Primes have exactly the same dynamics. All Prime Quintuplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number). In the hexagon, Prime Quintuplet are generated only at (6n -1)(6n+5). [n is a positive integer] When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quintuplet are 96/3 times of the 5th power distribution of primes, the frequency of occurrence of Prime Quintuplet is very equal to 0. However, it is not 0. Therefore, Prime Quintuplet continue to be generated. If Prime Quintuplet is finite, the Primes is finite. The probability of Prime Quintuplet 96/3 times of the 5th power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes. That is, Prime Quintuplet exist forever.
[318]
vixra:1911.0177 [
pdf]
Sexy Primes Conjecture
Sexy Primes Conjecture were prooved. Sexy Primes and Twin Primes and Cousin Primes have exactly the same dynamics. All Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number). In the hexagon, Sexy Primes are generated only at (6n+1)(6n -1). [n is a positive integer] When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Sexy Primes are 8/3 times the square of the distribution of primes, the frequency of occurrence of Sexy Primes is very equal to 0. However, it is not 0. Therefore, Sexy Primes continue to be generated. If Sexy Primes is finite, the Primes is finite. Because, Sexy Primes are 8/3 times the square of the distribution of primes. This is contradictory. Since there are an infinite of Primes. That is, Sexy Primes exist forever.
[319]
vixra:1911.0144 [
pdf]
Prime Quadruplet Conjecture
Prime Quadruplet and Twin Primes have exactly the same dynamics. All Prime Quadruplet are executed in hexagonal circulation. It does not change in a huge number (forever huge number). In the hexagon, Prime Quadruplet are generated only at (6n -1)(6n+5). [n is a positive integer] When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Prime Quadruplet are 16/3 of the fourth power distribution of primes, the frequency of occurrence of Prime Quadruplet is very equal to 0. However, it is not 0. Therefore, Cousin Primes continue to be generated. If Prime Quadruplet is finite, the Primes is finite. The probability of Prime Quadruplet 16/3 of the fourth power probability of appearance of the Prime. This is contradictory. Because there are an infinite of Primes. That is, Prime Quadruplet exist forever.
[320]
vixra:1911.0083 [
pdf]
Cousin Primes Conjecture
Cousin Primes Conjecture were performed using WolframAlpha and Wolfram cloud from the beginning this time, as in the case of the twin primes that we did the other day. Cousin Primes and Twin Primes have exactly the same dynamics. All Cousin Primes are executed in hexagonal circulation. It does not change in a huge number (forever huge number). In the hexagon, Cousin Primes are generated only at (6n+1)(6n+5). [n is a positive integer] When the number grows to the limit, the denominator of the expression becomes very large, and primes occur very rarely, but since Cousin Primes are 4/3 times the square of the distribution of primes, the frequency of occurrence of Cousin Primes is very equal to 0. However, it is not 0. Therefore, Cousin Primes continue to be generated. That is, Cousin Primes exist forever.
[321]
vixra:1911.0002 [
pdf]
In Twin prime Conjecture Constance 4/3
I proved the Twin Prime Conjecture. However, a new problem of mystery with a Constance 4/3 occurred. I have studied this in various ways, but I don't know.
[322]
vixra:1910.0551 [
pdf]
Using Decimals to Prove Zeta(n >= 2) is Irrational
With a strange and ironic twist an open number theory problem, show Zeta(n) is irrational for natural numbers greater than or equal to 2, is solved with the easiest of number theory concepts: the rules of representing fractions with decimals.
[323]
vixra:1910.0494 [
pdf]
The Proof of Goldbach’s Conjecture
Since the set of AS(+) and AS(×) is a bijective function, we use the improved the theorem of asymptotic density to prove that there exist prodcut of two odd primes in any AS(×). At the same time, in any AS(+), the sum of two odd primes can be obtained.
[324]
vixra:1910.0366 [
pdf]
A Complete Proof of Beal's Conjecture
In 1997, Andrew Beal announced the following conjecture: textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending on $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p$ and a parameter $theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or do have not a common factor. Three numerical examples are given.
[325]
vixra:1910.0281 [
pdf]
A New Look at Potential vs. Actual Infinity
The {\it technique} of classical mathematics involves only potential infinity, i.e. infinity is understood only as a limit, and, as a rule, legitimacy of every limit is thoroughly investigated. However, {\it the basis} of classical mathematics does involve actual infinity: the infinite ring of integers $Z$ is the starting point for constructing infinite sets with different cardinalities, and, even in standard textbooks on classical mathematics, it is not even posed a problem whether $Z$ can be treated as a limit of finite sets. On the other hand, finite mathematics starts from the ring $R_p=(0,1,...p-1)$ (where all operations are modulo $p$) and the theory deals only with a finite number of elements. We give a direct proof that $Z$ can be treated as a limit of $R_p$ when $p\to\infty$, and the proof does not involve actual infinity. Then we explain that, as a consequence, finite mathematics is more fundamental than classical one.
[326]
vixra:1910.0239 [
pdf]
Inequality in the Universe, Imaginary Numbers and a Brief Solution to P=NP? Problem
While I was working about some basic physical phenomena, I discovered some geometric relations that also interest mathematics. In this work, I applied the rules I have been proven to P=NP? problem over impossibility of perpendicularity in the universe. It also brings out extremely interesting results out like imaginary numbers which are known as real numbers currently. Also it seems that Euclidean Geometry is impossible. The actual geometry is Riemann Geometry and complex numbers are real.
[327]
vixra:1910.0081 [
pdf]
Twin Prime Conjecture (New Edition)
I proved the Twin Prime Conjecture. All Twin Primes are executed in hexadecimal notation. It does not change in a huge number (forever huge number). In the hexagon, prime numbers are generated only at [6n -1] [6n+1]. (n is a positive integer) The probability that a twin prime will occur is 6/5 times the square of the probability that a prime will occur. If the number is very large, the probability of generating a prime number is low, but since the prime number exists forever, the probability of generating a twin prime number is very low, but a twin prime number is produced. That is, twin primes exist forever.
[328]
vixra:1910.0077 [
pdf]
Grimm's Conjecture
The collection of the consecutive composite integers is the composite connected, and no pair of its distinct integers may be generated by a single prime number. Composite connectedness implies the two-primes rule and the singularity propagation/breaking rule.Failure of the singularity propagation proves the Gremm's Conjecture.
[329]
vixra:1909.0653 [
pdf]
ζ(4), ζ(6).......ζ(80), ζ(82) Are Irrational Number
ζ(4), ζ(6).......ζ(80), ζ(82) considered. From these equations, it can be said that ζ(4),ζ(6).......ζ(80),ζ(82) are irrational numbers. ζ(84),ζ(86) etc. can also be expressed by these equations. Because I use π2, these are to be irrational numbers. The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of ζ(2n) is multiplied by π2.
[330]
vixra:1909.0515 [
pdf]
The Requirements on the Non-trivial Roots of the Riemann Zeta via the Dirichlet Eta Sum
An explanation of the Riemann Hypothesis is given in sections, using the well known Dirichlet Eta sum equivalence, beginning with a brief history of the paper and a statement of the problem. The next 3 sections dissect the complex Eta sum into 8 real valued sums and 2 constants. Parts 6 and 8 explain a recursive relationship between the sums and constants, via 2 systems of 2 equations, while parts 7 and 9 explain the conditions generated from both systems. Finally, section 10 concludes the explanation in terms of the original inputs of the Dirichlet Eta sum, proves Riemann's suspicion, and it shows that the only possible solution for the real portion of the complex input, commonly labeled a, is that it must equal 1/2 and only 1/2.
[331]
vixra:1909.0473 [
pdf]
Formula of ζ Even-Numbers
I published the odd value formula for ζ, but I realized that this was true even when it was even. Therefore, it will be announced.
[332]
vixra:1909.0461 [
pdf]
Fibonacci's Answer to Primality Testing?
In this paper, we consider various approaches to primality testing and then ask whether an effective deterministic test for prime numbers can be found in the Fibonacci numbers.
[333]
vixra:1909.0385 [
pdf]
Formula of ζ Odd-Numbers
I tried to find a new expression for zeta odd-numbers. It may be a new expression and will be published here. The correctness of this formula was confirmed by WolframAlpha to be numerically com- pletely correct.
[334]
vixra:1909.0384 [
pdf]
ζ(4), ζ(6).......ζ(108), ζ(110) Are Irrational Number
ζ(4), ζ(6).......ζ(108), ζ(110) considered. From these equations, it can be said that ζ(4),ζ(6).......ζ(108),ζ(110) are irrational numbers. ζ(112),ζ(114) etc. can also be expressed by these equations. Because I use π2, these are to be irrational numbers. The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of ζ(2n) is multiplied by π2.
[335]
vixra:1909.0334 [
pdf]
The Characteristics of Primes
The prime numbers has very irregular pattern. The problem of finding pattern in the prime numbers is the long-open problem in mathematics. In this paper, we try to solve the problem axiomatically. We propose some natural properties of prime numbers.
[336]
vixra:1909.0315 [
pdf]
ζ(5), ζ(7)..........ζ(331), ζ(333) Are Irrational Number
Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers. ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement. This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.
[337]
vixra:1909.0059 [
pdf]
If Riemann’s Zeta Function is True, it Contradicts Zeta’s Dirichlet Series, Causing "Explosion". If it is False, it Causes Unsoundness.
Riemann's "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane $s \in \mathbb{C}$, $\text{Re}(s)\le1$, (except at $s=1$). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function cannot be both convergent and divergent at any domain value. In other mathematics conjectures and assumed-proven theorems, and in physics, the Riemann Zeta function (or the class of $L$-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle's Laws of Identity, Non-Contradiction, and Excluded Middle. The of Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo-Fraenkel set theory (which was designed to avoid paradoxes). If Riemann's definition of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann's Zeta is false, it renders unsound all theorems and conjectures that falsely assume that it is true. Riemann's Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy's integral theorem.
[338]
vixra:1908.0427 [
pdf]
The Riemann Hypothesis Proof
We take the integral representation of the Riemann Zeta Function over entire complex plane, except for a pole at 1. Later we draw an equivalent to the Riemann Hypothesis by studying its monotonicity properties.
[339]
vixra:1908.0420 [
pdf]
A Final Proof of The abc Conjecture
In this paper, we consider the abc conjecture. As the conjecture c<rad^2(abc) is less open, we give firstly the proof of a modified conjecture that is c<2rad^2(abc). The factor 2 is important for the proof of the new conjecture that represents the key of the proof of the main conjecture. Secondly, the proof of the abc conjecture is given for \epsilon \geq 1, then for \epsilon \in ]0,1[. We choose the constant K(\epsion) as K(\epsilon)=2e^{\frac{1}{\epsilon^2} } for $\epsilon \geq 1 and K(\epsilon)=e^{\frac{1}{\epsilon^2}} for \epsilon \in ]0,1[. Some numerical examples are presented.
[340]
vixra:1908.0307 [
pdf]
On Certain Pi_{q}-Identities of W. Gosper
In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Pi_{q}-identities. As a consequence, a q-identity involving Pi_{q} and Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper.
[341]
vixra:1907.0463 [
pdf]
Analytic Continuation of the Zeta Function Violates the Law of Non-Contradiction (LNC)
The Dirichlet series of the Zeta function was long ago proven to be divergent throughout half-plane Re(s) =< 1. If also Riemann's proposition is true, that there exists an "expression" of the Zeta function that is convergent at all values of s (except at s = 1), then the Zeta function is both divergent and convergent throughout half-plane Re(s) =< 1 (except at s = 1). This result violates all three of Aristotle's "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non-Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion": Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann's analytic continuation of the Zeta function violates Cauchy's integral theorem, providing another proof of the invalidity of analytic continuation of the Zeta function. Also, Riemann's Zeta function is one of the L-functions, which are all invalid, because they are generalizations of the invalid analytic continuation of the Zeta function. This result renders unsound all theorems (e.g. Modularity, Fermat's last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann's Zeta function) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar's 3VL, Priest's LP).
[342]
vixra:1907.0437 [
pdf]
Values of the Riemann Zeta Function by Means of Division by Zero Calculus
In this paper, we will give the values of the Riemann zeta function for any positive integers by means of the division by zero calculus. Zero, division by zero, division by zero calculus, $0/0=1/0=z/0=\tan(\pi/2) = \log 0 =0 $, Laurent expansion, Riemann zeta function, Gamma function, Psi function, Digamma function.
[343]
vixra:1907.0206 [
pdf]
On Non-Trivial Zero Point
In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive. We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5. This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value. We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.
[344]
vixra:1907.0037 [
pdf]
Disproof of the Riemann Hypothesis
In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained. It is impossible to make c = 0.5 exactly like this. c can only be 0.5 and its edge. It is considered that “when the imaginary value increases to infinity, the denominator of the number becomes infinity and shifts from 0.5 to 0”.
[345]
vixra:1906.0463 [
pdf]
Finding The Hamiltonian
We first find a Hamiltonian H that has the Hurwitz zeta functions ζ(s,x) as eigenfunctions. Then we continue constructing an operator G that is self-adjoint, with appropriate boundary conditions. We will find that the ζ(s,x)-functions do not meet these boundary conditions, except for the ones where s is a nontrivial zero of the Riemann zeta, with the real part of s being greater than 1/2. Finally, we find that these exceptional functions cannot exist, proving the Riemann hypothesis, that all nontrivial zeros have real part equal to 1/2.
[346]
vixra:1906.0377 [
pdf]
Zero Points of Riemann Zeta Function
In this article, we assume that the Riemann Zeta Function equals to the Euler product at the non zero points of the Riemann Zeta function. From this assumption we can prove that there are no zero points of Riemann Zeta function, ς(s) in Re(s) > 1/2. We applied proof by contradiction.
[347]
vixra:1906.0374 [
pdf]
A Simple Proof for Catalan's Conjecture
Catalan's Conjecture was first made by Belgian mathematician Eugène Charles Catalan in 1844, and states that 8 and 9 (2^3 and 3^2) are the only consecutive powers, excluding 0 and 1. That is to say, that the only solution in the natural numbers of a^x - b^y=1 for a,b,x,y > 1 is a=3, x=2, b=2, y=3. In other words, Catalan conjectured that 3^2-2^3=1 is the only nontrivial solution. It was finally proved in 2002 by number theorist Preda Mihailescu making extensive use of the theory of cyclotomic fields and Galois modules.
[348]
vixra:1906.0318 [
pdf]
Asymptotic Closed-form Nth Zero Formula for Riemann Zeta Function
Assuming the Riemann Hypothesis to be true, we propose an asymptotic and closed-form formula to find the imaginary part for non-trivial zeros of the Riemann Zeta Function.
[349]
vixra:1906.0243 [
pdf]
Speed and Measure Theorems Related to the Lonely Runner Conjecture
We prove an important new result on this problem: Given any epsilon > 0 and k >= 5, and given any set of speeds s_1 < s_2 < ... < s_k, there is a set of speeds v_1 < v_2 < ... < v_k for which the lonely runner conjecture is true and for which |s_i - v_i| < epsilon. We also prove some measure theorems.
[350]
vixra:1906.0199 [
pdf]
A Concise Proof for Beal's Conjecture
In this paper, we show how a^x - b^y can be expressed as a binomial expansion (to an indeterminate power, z, and use it as the basis for a proof for the Beal Conjecture.
[351]
vixra:1906.0025 [
pdf]
A New Proof of the ABC Conjecture
In this paper, using the recent result that $c<rad(abc)^2$, we will give the proof of the $abc$ conjecture for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\frac{1}{\epsilon^2} $. Some numerical examples are presented.
[352]
vixra:1905.0614 [
pdf]
Nature Works the Way Number Works
Based on Eulers formula a concept of dually unit or d-unit circle is discovered. Continuing with, Riemann hypothesis is proved from different angles, Zeta values are renormalised to remove the poles of Zeta function and relationships between numbers and primes is discovered. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be defined such a way that it eases the complex logarithm without needing branch cuts. Pi can also be a base to natural logarithm and complement complex logarithm.Grand integrated scale is discovered which can reconcile the scale difference between very big and very small. Complex constants derived from complex logarithm following Goldbach partition theorem and Eulers Sum to product and product to unity can explain lot of mysteries in the universe.
[353]
vixra:1905.0546 [
pdf]
Consideration of the Riemann Hypothesis
I considered Riemann’s hypothesis. At first, the purpose was to prove, but can not to prove. It is written in the middle of the proof, but it can not been proved at all. (The calculation formula is also written, but the real value 0.5 was not shown at all) The non-trivial zero values match perfectly in the formula of this paper. However, the formula did not reach the real value 0.5. In this case, it only reaches the pole near the real value 0.5.
[354]
vixra:1905.0468 [
pdf]
Crazy proof of Fermat's Last Theorem
This paper magically shows very interesting and simple proof of Fermat’s Last Theorem. The proof identifies sufficient derivations of equations that holds the statement true and describes contradictions on them to satisfy the theorem. If Fermat had proof, his proof is most probably similar to this one. The proof does not require any higher field of mathematics and it can be understood in high school level of mathematics. It uses only modular arithmetic, factorization and some logical statements.
[355]
vixra:1905.0269 [
pdf]
Zeros of Gamma
160 years ago that in the complex analysis a hypothesis was raised, which was used in principle to demonstrate a theory about prime numbers, but, without any proof; with the passing Over the years, this hypothesis has become very important, since it has multiple applications to physics, to number theory, statistics, among others In this article I present a demonstration that I consider is the one that has been dodging all this time.
[356]
vixra:1905.0041 [
pdf]
A Final Tentative of The Proof of The ABC Conjecture - Case c=a+1
In this paper, we consider the abc conjecture in the case c=a+1. Firstly, we give the proof of the first conjecture that c<rad^2(ac) using a polynomial function. It is the key of the proof of the abc conjecture. Secondly, the proof of the abc conjecture is given for \epsilon >1, then for \epsilon \in ]0,1[ for the two cases: c<rad(ac) and c> rad(ac). We choose the constant K(\epsilon) as K(\epsilon)=e^{\frac{1}{\epsilon^2}). A numerical example is presented.
[357]
vixra:1905.0021 [
pdf]
Weights at the Gym and the Irrationality of Zeta(2)
This is an easy approach to proving zeta(2) is irrational. The reasoning is by analogy with gym weights that are rational proportions of a unit. Sometimes the sum of such weights is expressible as a multiple of a single term in the sum and sometimes it isn't. The partials of zeta(2) are of the latter type. We use a result of real analysis and this fact to show the infinite sum has this same property and hence is irrational.
[358]
vixra:1904.0489 [
pdf]
Sums of Powers of the Terms of Lucas Sequences with Indices in Arithmetic Progression
We evaluate the sums $\sum_{j=0}^k{u_{rj+s}^{2n}\,z^j}$, $\sum_{j=0}^k{u_{rj+s}^{2n-1}\,z^j}$ and $\sum_{j=0}^k{v_{rj+s}^{n}\,z^j}$, where $r$, $s$ and $k$ are any integers, $n$ is any nonnegative integer, $z$ is arbitrary and $(u_n)$ and $(v_n)$ are the Lucas sequences of the first kind and of the second kind, respectively. As natural consequences we obtain explicit forms of the generating functions for the powers of the terms of Lucas sequences with indices in arithmetic progression. This paper therefore extends the results of P.~Sta\u nic\u a who evaluated $\sum_{j=0}^k{u_{j}^{2n}\,z^j}$ and $\sum_{j=0}^k{u_{j}^{2n-1}\,z^j}$; and those of B. S. Popov who obtained generating functions for the powers of these sequences.
[359]
vixra:1904.0376 [
pdf]
Nature Works the Way Number Works
Based on Euler ’s formula a concept of duality unit or dunit circle is discovered. Continuing with Riemann hypothesis is proved from different angles, zeta values are renormalised to remove the poles of zeta function and discover relationships between numbers and primes. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be defined such a way that it eases the complex logarithm and accounts for the scale difference between very big and very small. Pi can also be a base to natural logarithm and complement the scale gap. 96 complex constants derived from complex logarithm can explain everything in the universe.
[360]
vixra:1904.0146 [
pdf]
A Tentative of The Proof of The ABC Conjecture - Case c=a+1
In this paper, we consider the $abc$ conjecture in the case $c=a+1$. Firstly, we give the proof of the first conjecture that $c<rad^2(ac)$ using the polynomial functions. It is the key of the proof of the $abc$ conjecture. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$ for the two cases: $ c\leq rad(ac)$ and $c> rad(ac)$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\ds \left(\frac{1}{\epsilon^2} \right)}$. A numerical example is presented.}
[361]
vixra:1903.0503 [
pdf]
Extending an Irrationality Proof of Sondow: from e to Zeta(n)
We modify Sondow's geometric proof of the irrationality of e. The modification uses sector areas on circles, rather than closed intervals. Using this circular version of Sondow's proof, we see a way to understand the irrationality of a series. We evolve the idea of proving all possible rational value convergence points of a series are excluded because all partials are not expressible as fractions with the denominators of their terms. If such fractions cover the rationals, then the series should be irrational. Both the irrationality of e and that of zeta(n>=2) are proven using these criteria: the terms cover the rationals and the partials escape the terms.
[362]
vixra:1903.0157 [
pdf]
Consideration of Riemann Hypothesis 43 Counterexamples
I also found a zero point which seems to deviate from 0.5. I thought that the zero point outside 0.5 can not be found very easily in the area which can not be shown in the figure, but this area can not be represented in the figure but can be found one after another. It is completely unknown whether this axis is distorted in the 0.5 axis or just by coincidence. The number of zero points in the area that can not be shown in the figure is now 43. No matter how you looked it was not found in other areas. It seemed that there is no other way to interpret this axis as 0.5 axis is distorted in this area. Somewhere on the net there is a memory that reads the mathematician's view that "there are countless zero points in the vicinity of 0.5 on high area". We are reporting that the zero point search of the high-value area of the imaginary part which was giving up as it is no longer possible with the supercomputer is no longer possible, is reported. 43 zero-point searches in the high-value area of the imaginary part are thus successful. This means that the zero point search in the high-value area of the imaginary part has succeeded in the 43. We will also write 43 zero point searches of the successful high-value area of the imaginary part. There are many counterexamples far beyond 0.5, which is far beyond the limit, but the computer can not calculate it. Moreover, I believe that it can only be confirmed on supercomputer whether this is really counterexample. In addition, it is necessary to make corrections in the supercomputer.
[363]
vixra:1903.0059 [
pdf]
Are Imaginary Numbers Rooted in an Asymmetric Number System? The Alternative is a Symmetric Number System!
In this paper, we point out an interesting asymmetry in the rules of fundamental mathematics between positive and negative numbers. Further, we show that there exists an alternative numerical system that is basically identical to today’s system, but where positive numbers dominate over negative numbers. This is like a mirror symmetry of the existing number system. The asymmetry in both of these systems leads to imaginary and complex numbers. We also suggest an alternative number system with perfectly symmetrical rules – that is, where there is no dominance of negative numbers over positive numbers, or vice versa, and where imaginary and complex numbers are no longer needed. This number system seems to be superior to other numerical systems, as it brings simplicity and logic back to areas that have been dominated by complex rules for much of the history of mathematics. We also briefly discuss how the Riemann hypothesis may be linked to the asymmetry in the current number system. The foundation rules of a number system can, in general, not be proven incorrect or correct inside the number system itself. However, the ultimate goal of a number system is, in our view, to be able to describe nature accurately. The optimal number system should therefore be developed with feedback from nature. If nature, at a very fundamental level, is ruled by symmetry, then a symmetric number system should make it easier to understand nature than a asymmetric number system would. We hypothesize that a symmetric number system may thus be better suited to describing nature. Such a number system should be able to get rid of imaginary numbers in space-time and quantum mechanics, for example, two areas of physics that to this day are clouded in mystery.
[364]
vixra:1902.0235 [
pdf]
The Proof of The ABC Conjecture - Part I: The Case c=a+1
In this paper, we consider the abc conjecture in the case c=a+1. Firstly, we give the proof of the first conjecture that c<rad*2(ac). It is the key of the proof of the abc conjecture. Secondly, the proof of the abc conjecture is given for \epsilon \geq 1, then for \epsilon \in ]0,1[ for the two cases: c\leq rad(ac) and c> rad(ac). We choose the constant K(\epsilon) as K(\epsilon)=e^{\left(\frac{1}{\ep*2} \right)}. A numerical example is presented.
[365]
vixra:1902.0147 [
pdf]
Definitive Proof of the Near-Square Prime Conjecture, Landau’s Fourth Problem
The Near-Square Prime conjecture, states that there are an infinite number of prime numbers of the form x^2 + 1. In this paper, a function was derived that determines the number of prime numbers of the form x^2 + 1 that are less than n^2 + 1 for large values of n. Then by mathematical induction, it is proven that as the value of n goes to infinity, the function goes to infinity, thus proving the Near-Square Prime conjecture.
[366]
vixra:1902.0040 [
pdf]
A Complete Proof of the abc Conjecture: The End of The Mystery
In this paper, we consider the abc conjecture. Firstly, we give a proof of a first conjecture that c<rad*2(abc). It is the key of the proof of the abc conjecture. Secondly, a proof of the abc is given for \ep \geq 1, then for \ep \in ]0,1[ for the two cases: c\leq rad(abc) and c>rad(abc). We choose the constant K(\ep) as K(\ep)=6*{1+\ep}e*{\left(\frac{1}{\ep*2}-\ep \right)}. Five numerical examples are presented.
[367]
vixra:1901.0436 [
pdf]
Definitive Proof of Legendre's Conjecture
Legendre's conjecture, states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that accurately determines the number of prime numbers less than n for large values of n. Then, using this equation, it was proven by induction that there is at least one prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture for sufficiently large values n. The error between the derived equation and the actual number of prime numbers less than n was empirically proven to be very small (0.291% at n = 50,000), and it was proven that the size of the error declines as n increases, thus validating the proof.
[368]
vixra:1901.0430 [
pdf]
A Note About the Abc Conjecture a Proof of the Conjecture: C<rad*2(abc)
In this paper, we consider the abc conjecture, then we give a proof of the conjecture c<rad^2(abc) that it will be the key to the proof of the abc conjecture.
[369]
vixra:1901.0108 [
pdf]
Assuming ABC Conjecture is True Implies Beal Conjecture is True
In this paper, we assume that the ABC conjecture is true, then we give a proof that Beal conjecture is true. We consider that Beal conjecture is false then we arrive to a contradiction. We deduce that the Beal conjecture is true.
[370]
vixra:1901.0101 [
pdf]
A Resolution Of The Brocard-Ramanujan Problem
We identify equivalent restatements of the Brocard-Ramanujan diophantine equation, $(n! + 1) = m^2$; and employing the properties and implications of these equivalencies, prove that for all $n > 7$, there are no values of $n$ for which $(n! + 1)$ can be a perfect square.
[371]
vixra:1901.0007 [
pdf]
On the Prime Decomposition of Integers of the Form (Z^n-Y^n)/(z-y)
In this work, the author shows a sufficient and necessary condition for an integer of the form (z^n-y^n)/(z-y) to be divisible by some perfect mth power p^m,where p is an odd prime and m is a positive integer. A constructive method of this type of integers is explained with details and examples. Links beetween the main result and known ideas such as Fermat’s last theorem, Goor-maghtigh conjecture and Mersenne numbers are discussed. Other relatedideas, examples and applications are provided.
[372]
vixra:1812.0340 [
pdf]
Modular Logarithms Unequal
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction the result that two finite numbers is with unequal logarithms in a corresponding module is proven, which result is applied to solving a kind of high degree diophantine equation.
[373]
vixra:1812.0312 [
pdf]
Another Proof for Catalan's Conjecture
In 2002 Preda Mihailescu used the theory of cyclotomic fields and Galois modules to prove Catalan's Conjecture. In this short paper, we give a very simple proof. We first prove that no solutions exist for a^x-b^y=1 for a,b>0 and x,y>2. Then we prove that when x=2 the only solution for a is a=3 and the only solution for y is y=3.
[374]
vixra:1812.0305 [
pdf]
On the Distribution of Prime Numbers
In this paper it is proposed and proved an exact formula for the prime-counting function, finding an expression of Legendre's formula. As corollaries, they are proved some important conjectures regarding prime numbers distribution.
[375]
vixra:1812.0208 [
pdf]
Definitive Proof of the Twin-Prime Conjecture
A twin prime is defined as a pair of prime numbers (p1,p2) such that p1 + 2 = p2. The Twin Prime Conjecture states that there are an infinite number of twin primes. A more general conjecture by de Polignac states that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case where k = 1 is the Twin Prime Conjecture. In this document, a function is derived that corresponds to the number of twin primes less than n for large values of n. Then by proof by induction, it is shown that as n increases indefinitely, the function also increases indefinitely thus proving the Twin Prime Conjecture. Using this same methodology, the de Polignac Conjecture is also shown to be true.
[376]
vixra:1812.0182 [
pdf]
A Proposed Proof of The ABC Conjecture
In this paper, from a,b,c positive integers relatively prime with c=a+b, we consider a bounded of c depending of a,b. Then we do a choice of K(\epsilon) and finally we obtain that the ABC conjecture is true. Four numerical examples confirm our proof.
[377]
vixra:1812.0154 [
pdf]
L' Attraction Des Nombres Par la Force Syracusienne}
I study in which cases $ x \in \mathbb{N}^*$ and $1 \in \mathcal{O}_S (x)= \{ S^n(x), n \in \mathbb{N}^* \} $ where $ \mathcal{O}_S (x)$ is the orbit of the function S defined on $\mathbb{R}^+$ by $S(x)= \frac{x}{2} + (\frac{q-1}{2} x+\frac{1}{2}) sin^2(x\frac{\pi}{2})$ , $ q \in 2\mathbb{N}^*+1$. And I deduce the proof of the Syracuse conjecture.
[378]
vixra:1812.0107 [
pdf]
A Note About the Determination of The Integer Coordinates of An Elliptic Curve: Part I
In this paper, we give the elliptic curve (E) given by the equation: y^2=x^3+px+q with $p,q \in Z$ not null simultaneous. We study a part of the conditions verified by $(p,q)$ so that it exists (x,y) \in Z^2 the coordinates of a point of the elliptic curve (E) given by the equation above.
[379]
vixra:1812.0020 [
pdf]
A Complete Proof of the ABC Conjecture
In this paper, we assume that Beal conjecture is true, we give a complete proof of the ABC conjecture. We consider that Beal conjecture is false $\Longrightarrow$ we arrive that the ABC conjecture is false. Then taking the negation of the last statement, we obtain: ABC conjecture is true $\Longrightarrow$ Beal conjecture is true. But, if the Beal conjecture is true, then we deduce that the ABC conjecture is true
[380]
vixra:1812.0018 [
pdf]
Zeros of the Riemann Zeta Function can be Found Arbitrary Close to the Line \Re(s) =1
In this paper, not only did we disprove the Riemann Hypothesis (RH) but we also showed that zeros of the Riemann zeta function $\zeta (s)$ can be found arbitrary close to the line $\Re (s) =1$. Our method to reach this conclusion is based on analyzing the fine behavior of the partial sum of the Dirichlet series with the Mobius function $M (s) = \sum_n \mu (n) /n^s$ defined over $p_r$ rough numbers (i.e. numbers that have only prime factors greater than or equal to $p_r$). Two methods to analyze the partial sum fine behavior are presented and compared. The first one is based on establishing a connection between the Dirichlet series with the Mobius function $M (s) $ and a functional representation of the zeta function $\zeta (s)$ in terms of its partial Euler product. Complex analysis methods (specifically, Fourier and Laplace transforms) were then used to analyze the fine behavior of partial sum of the Dirichlet series. The second method to estimate the fine behavior of partial sum was based on integration methods to add the different co-prime partial sum terms with prime numbers greater than or equal to $p_r$. Comparing the results of these two methods leads to a contradiction when we assume that $\zeta (s)$ has no zeros for $\Re (s) > c$ and $c <1$.
[381]
vixra:1811.0250 [
pdf]
A Proof Minus Epsilon of the Abc Conjecture
In this paper, we give a proof minus $\epsilon$ of the $ABC$ conjecture, considering that Beal conjecture is true. Some conditions are proposed for the proof, perhaps it needs some justifications that is why I give the title of the paper " a proof minus $\epsilon$ of the $ABC$ conjecture".
[382]
vixra:1811.0112 [
pdf]
A New Proof of the Strong Goldbach Conjecture
The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture.\\ The only best known result so far is that of Chen [3], proving that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to $4.10^{18}$ in 2014 , J\"erg [4] and Tom\'as [5]. In this paper, we prove that the conjecture is true for all even integers greater than 8.
[383]
vixra:1810.0459 [
pdf]
Twin Primes
This paper gives us an application of Eratosthenes sieve to distribution mean distance between primes using first and upper orders of Gauss integral log- arithm Li(x).We define function Υ in section 5. Sections 1 − 4 give us an introduction to the terminology and a clarification on Υ terms. Section 6 reassumes foregoing explanations and gives us two theorems using first and upper integral logarithm orders.
[384]
vixra:1810.0423 [
pdf]
The Formula of Zeta Odd Number
I calculated ζ (3),ζ(5). ζ (7),ζ(9)……… ζ (23). And the formula indicated. For example, in ζ (3) For example, in ζ (5) And ultimately the following formula is required n and m are positive integer.
[385]
vixra:1810.0335 [
pdf]
Using Cantor's Diagonal Method to Show Zeta(2) is Irrational
We look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We also puzzle out why the convergence of the constructed number, its value, is of no concern. We next review general properties of decimals and prove the existence of an irrational number with a modified version of Cantor's diagonal method. Finally, we show, with yet another modification of the method, that Zeta(2) is irrational.
[386]
vixra:1810.0175 [
pdf]
New Abelian Groups for Primes of Type 4K-1 and 4K+1.
p is prime.The article describes the new Abelian groups of type p=4k+1 and p = 4k-1, for which a theorem similar to the Fermat's little theorem applies. The multiplicative group (Z/pZ)* in some sense similar to the Abelian group of type p = 4k+1. Abelian group of type p = 4k-1 is a different structure compared to group (Z/pZ)*. This fact is used for the primality test of integer N = 4k-1. The primality test was veried up to N = 2^(64).
[387]
vixra:1810.0046 [
pdf]
Riemann Conjecture Proof
The main contribution of this paper is to achieve the proof of Riemann hypothesis. The key idea is based on new formulation of the problem $$\zeta(s)=\zeta(1-s) \Leftrightarrow re(s)=\frac{1}{2}$$. This proof is considered as a great discovery in mathematic.
[388]
vixra:1809.0351 [
pdf]
The Cordiality for the Conjunction of Two Paths
Abstract A graph is called cordial if it has a 0 - 1 labeling such that the number of vertices (edges) labeled with ones and zeros dier by at most one. The conjunction of two graphs (V1;E1) and (V2;E2) is the graph G = (V;E), where V = V1 x V2 and u = (a1; a2), v = (b1; b2) are two vertices, then uv belongs to E if aibi belongs to Ei for i = 1 or 2. In this paper, we present necessary and sucient condition for cordial labeling for the conjunction of two paths, denoted by Pn ^ Pm. Also, we drive an algorithm to generate cordial labeling for the conjunction Pn ^ Pm.
[389]
vixra:1809.0086 [
pdf]
An Identity for Horadam Sequences
We derive an identity connecting any two Horadam sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed.
[390]
vixra:1809.0059 [
pdf]
A Proof of Benfords Law in Geometric Series
We show in this paper another proof of Benford’s Law. The idea starts with the problem of to find the first digit of a power. Then we deduced a function to calculate the first digit of any power a j called L f function. The theorem 1.2 its a consequence of the periodicity of the $L_f$ function.
[391]
vixra:1808.0567 [
pdf]
A Proof For Beal's Conjecture
In the first part of this paper, we show how a^x - b^y can be expressed as a new non-standard binomial formula (to an indeterminate power, n). In the second part, by fixing n to the value of z we compare this binomial formula to the standard binomial formula for c^z to prove the Beal Conjecture.
[392]
vixra:1808.0531 [
pdf]
Goldbach's Conjecture
I proved the Goldbach's conjecture. Even numbers are prime numbers and prime numbers added, but it has not been proven yet whether it can be true even for a huge number (forever huge number). All prime numbers are included in (6n - 1) or (6n + 1) except 2 and 3 (n is a positive integer). All numbers are executed in hexadecimal notation. This does not change even in a huge number (forever huge number). 2 (6n + 2), 4 (6n - 2), 6 (6n) in the figure are even numbers. 1 (6n + 1), 3 (6n + 3), 5 (6n - 1) are odd numbers.
[393]
vixra:1808.0509 [
pdf]
Gamma is Irrational
We introduce an unaccustomed number system, H±, and show how it can be used to prove gamma is irrational. This number system consists of plus and minus multiplies of the terms of the harmonic series. Using some properties of ln, this system can depict the harmonic series and lim as n goes to infinity of ln n at the same time, giving gamma as an infinite decimal. The harmonic series converges to infinity so negative terms are forced. As all rationals can be given in H± without negative terms, it follows that must be irrational.
[394]
vixra:1807.0510 [
pdf]
On the Representation of Even Integers by the Sum of Prime Numbers
The main objective of this short note is prove that some statements concerning the representation of even integers by the sum of prime numbers are equivalent to some true trivial case. This implies that these statements are also true. The analysis is based on a new prime formula and some trigonometric expressions.
[395]
vixra:1806.0052 [
pdf]
Identities for Second Order Recurrence Sequences
We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a harmonized study of six well known integer sequences, namely the Fibonacci sequence, the sequence of Lucas numbers, the Jacobsthal sequence, the Jacobsthal-Lucas sequence, the Pell sequence and the Pell-Lucas sequence.
[396]
vixra:1806.0051 [
pdf]
Weighted Tribonacci sums
We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.
[397]
vixra:1805.0325 [
pdf]
Demonstration Riemann's Hypothesis
158 years ago that in the complex analysis a hypothesis was raised, which was used in principle to demonstrate a theory about prime numbers, but, without any proof; with the passing Over the years, this hypothesis has become very important, since it has multiple applications to physics, to number theory, statistics, among others In this article I present a demonstration that I consider is the one that has been dodging all this time.
[398]
vixra:1805.0230 [
pdf]
Solution of the Erdös-Moser Equation 1+2^p+3^p+...+(k)^p=(k+1)^p
The Erdös-Moser equation (EM equation), named after Paul Erdös and Leo Moser, has been studied by many number theorists throughout history since combines addition, powers and summation together. An open and very interesting conjecture of Erdös-Moser states that there is no other solution of the EM equation than trivial 1+2=3. Investigation of the properties and identities of the EM equation and ultimately prove the conjecture is the main purpose of this article.
[399]
vixra:1805.0185 [
pdf]
Revisit of Carmichael 1913 Work and an Elementary Approach for Fermat’s Last Theorem of Case I
We discuss an elementary approach to prove the first case of Fermat's last theorem (FLT). The essence of the proof is to notice that $a+b+c$ is of order $N^{\alpha}$ if $a^N+b^N+c^N=0$. To prove FLT, we first show that $\alpha$ can not be $2$; we then show that $\alpha$ can not be $3$, etc. While this is is the standard method of induction, we refer to it here as the ``infinite ascent'' technique, in contrast to Fermat's original ``infinite descent'' technique. A conjecture, first noted by Ribenboim is used.
[400]
vixra:1805.0152 [
pdf]
On Squarefree Values of some Univariate Polynomials.
We consider univariate Polynomials, P(s), of the form (a1 * s + b1)*...*(ak * s + bk), where a1,..,ak,b1,..,bk are natural numbers and the variable s is squarefree. We give an algorithm to calculate, for a arbitrary s, the probability that the value of P(s) is squarefree.
[401]
vixra:1804.0416 [
pdf]
Statistical Bias in the Distribution of Prime Pairs and Isolated Primes
Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $\pi^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towards nonsquarefree numbers, a phenomenon most likely previously unknown. For twins, this leads to nonsquarefree numbers gaining an excess of $1.2\%$ of the total number of twins. In the case of isolated primes, this excess for nonsquarefree numbers amounts to $0.4\%$ of the total number of such primes. The above numbers are for the first $10^{10}$ primes, with the bias showing a tendency to grow, at least for isolated primes.
[402]
vixra:1804.0385 [
pdf]
Q-Analogues for Ramanujan-Type Series
From a very-well-poised _{6}\phi_{5} series formula we deduce a general series expansion formula involving the q-gamma function. With this formula we can give q-analogues of many Ramanujan-type series.
[403]
vixra:1804.0376 [
pdf]
The Strong Goldbach Conjecture, Klein Bottle And Möbius Strip
This modest article shows the connection between the strong Goldbach conjecture and the topological properties of the Klein bottle and the Möbius strip. This connection is established by functions derived from the number of divisors of the two odd integers whose sum is an even number.
[404]
vixra:1804.0267 [
pdf]
Intuitive Explanation of the Riemann Hypothesis
Let \alpha be the unique \Gamma(2) invariant form on H with a pole of residue 1 at i\infty and one of residue -1 at 1. The ratio [\alpha: i\pi dtau] tends to 1 at the upper limit of [0,i\infty). Let \mu_{pm}:TxH->H be the action of multiplying by \sqrt{g} and {1\over{\sqrt{g}} for g in the connected real multiplicative group T. For each real c in (0,1) and each unitary character \omega, the form g^{2-2c}\omega(g)\mu_+^*(\alpha-\pi d\tau)\wedge \mu_-^*)\alpha-i\pi d\tau is exact if and only if \zeta(c+i\omega_0)=0 where \omega_0 is chosen such that \omega(g)=g^{i\omega_0}. The rate of change of the magnitude is given by an integral involving a unitary character. Conjecturally the rate seminegative on the region 0 The form descends to the real projective line, it is locally meromorphic there with one pole and integrates to \pi e^{i\pi ({3\over 2}s + 1}. The value \zeta(s)=0 if and only if the integral along the arc from 0 to \infty not passing 1 is zero. This implies the arc passing through 1 equals a residue. We begin to relate the equality with the condition Re(s)=1/2.
[405]
vixra:1803.0317 [
pdf]
Analysis of Riemann's Hypothesis
Let p(c,r,v)=e^{(c-1)(r+2v)} log({{\lambda(r+v)}\over{q(r+v)}}) log({{\lambda(v)}\over{q(v)}}), f(c,r)=\int_{-\infty}^\infty p(c,r,v)+p(c,-r,v) dv. Let c be a real number such that 0<c<1/2.<br> Suppose that <br><br> f(c,r)<0 and {{\partial}\over{\partial r}}f(c,r)>0 for all $r\ge 0$ while {{\partial}\over{\partial c}}f(c,r)<0 and {{\partial^2}\over {\partial c \partial r}}f(c,r)>0 for all r>0. <br><br> Then \zeta(c+i\omega) \ne 0 for all \omega.
[406]
vixra:1803.0289 [
pdf]
New Discovery on Goldbach Conjecture
In this paper we are going to give the proof of Goldbach conjecture by introducing a new lemma which implies Goldbach conjecture .By using Chebotarev-Artin theorem , Mertens formula and Poincare sieve we establish the lemma
[407]
vixra:1803.0219 [
pdf]
Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects
The existing definition of imaginary numbers is solely based on the fact that certain mathematical operation, square operation, would not yield certain type of outcome, negative numbers; hence such operational outcome could only be imagined to exist. Although complex numbers actually form the largest set of numbers, it appears that almost no thought has been given until now into the full extent of all possible types of imaginary numbers. A close look into what further non-existing numbers could be imagined help reveal that we could actually expand the set of imaginary numbers, redefine complex numbers, as well as define imaginary and complex mathematical objects other than merely numbers.
[408]
vixra:1803.0179 [
pdf]
Continuity, Non-Constant Rate of Ascent, & The Beal Conjecture
The Beal Conjecture considers positive integers A, B, and C having respective positive integer exponents X, Y, and Z all greater than 2, where bases A, B, and C must have a common prime factor. Taking the general form A^X + B^Y = C^Z, we explore a small opening in the conjecture through reformulation and substitution to create two new variables. One we call 'C dot' representing and replacing C and the other we call 'Z dot' representing and replacing Z. With this, we show that 'C dot' and 'Z dot' are separate continuous functions, with argument (A^X + B^Y), that achieve all positive integers during their continuous non-constant rates of infinite ascent. Possibilities for each base and exponent in the reformulated general equation A^X +B^Y = ('C dot')^('Z dot') are examined using a binary table along with analyzing user input restrictions and 'C dot' values relative to A and B. Lastly, an indirect proof is made, where conclusively we find the continuity theorem to hold over the conjecture.
[409]
vixra:1802.0269 [
pdf]
Riemann's Analytic Continuation of Zeta(s) Contradicts the Law of the Excluded Middle, and is Derived by Using Cauchy's Integral Theorem While Contradicting the Theorem's Prerequisites
The Law of the Excluded Middle holds that either a statement "X" or its opposite "not X" is true. In Boolean algebra form, Y = X XOR (not X). Riemann's analytic continuation of Zeta(s) contradicts the Law of the Excluded Middle, because the Dirichlet series Zeta(s) is proven divergent in the half-plane Re(s)<=1. Further inspection of the derivation of Riemann's analytic continuation of $\zeta(s)$ shows that it is wrongly based on the Cauchy integral theorem, and thus false.
[410]
vixra:1801.0416 [
pdf]
Developing a Phenomenon for Analytic Number Theory
A phenomenon is described for analytic number theory. The purpose is to coordinate number theory and to give it a specific goal of modeling the phenomenon.
[411]
vixra:1801.0187 [
pdf]
A Conjecture of Existence of Prime Numbers in Arithmetic Progressions
In this paper it is proposed and proved a conjecture of existence of a prime number on the arithmetic progression S_{a,b}=\left\{ ab+1,ab+2,ab+3,...,ab+(b-1)\right\} As corollaries of this proof, they are proved many classical prime number’s conjectures and theorems, but mainly Bertrand's theorem, and Oppermann's, Legendre’s, Brocard’s, and Andrica’s conjectures. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated a corollary which implies some advance on the conjecture of the existence of infinite prime numbers of the form n^{2}+1.
[412]
vixra:1801.0140 [
pdf]
Simple Proofs that Zeta(n>=2) Is Irrational
We prove that partial sums of $\zeta(n)-1=z_n$ are not given by any single decimal in a number base given by a denominator of their terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational, possible convergence points. The limit of the partials is $z_n$ and the limit of the exclusions leaves only irrational numbers. Thus $z_n$ is proven to be irrational. Alternative proofs of this same type are given.
[413]
vixra:1801.0138 [
pdf]
Natural Squarefree Numbers: Statistical Properties II
This paper is an appendix of Natural Squarefree Numbers: Statistical Properties [PR04]. In this appendix we calculate the probability of c is squarefree, where c=a*b, a is an element of the set X and b is an element of the set Y.
[414]
vixra:1712.0662 [
pdf]
The Chameleon Effect, the Binomial Theorem and Beal's Conjecture
In psychology, the Chameleon Effect describes how an animal's behaviour can adapt to, or mimic, its environment through non-conscious mimicry. In the first part of this paper, we show how $a^x - b^y$ can be expressed as a binomial expansion (with an upper index, $z$) that, like a chameleon, mimics a standard binomial formula (to the power $z$) without its own value changing even when $z$ itself changes. In the second part we will show how this leads to a proof for the Beal Conjecture. We finish by outlining how this method can be applied to a more generalised form of the equation.
[415]
vixra:1712.0641 [
pdf]
Alternate Proof for Zeta(n>1) is Irrational
This is an alternative proof that zeta(n>1) is irrational. It uses nested intervals and Cantor's Nested Interval Theorem. It is a follow up for the article Visualizing Zeta(n>1) and Proving its Irrationality.
[416]
vixra:1712.0488 [
pdf]
Brief Solutions to Collatz Problem, Goldbach Conjecture and Twin Primes
I published some solutions a time ago to Goldbach Conjecture, Collatz Problem and Twin Primes; but I noticed that there were some serious logic voids to explain the problems. After that I made some corrections in my another article; but still there were some mistakes. Even so, I can say it easily that here I brought exact solutions for them out by new methods back to the drawing board.
[417]
vixra:1712.0441 [
pdf]
Natural Squarefree Numbers: Statistical Properties.
n this paper we calculate for various sets X (some subsets of the natural numbers) the probability of an element a of X is also squarefree. Furthermore we calculate the probability of c is squarefree, where c=a+b, a is an element of the set X and b is an element of the set Y.
[418]
vixra:1712.0384 [
pdf]
Discovering and Proving that Pi is Irrational, 2nd Edition
Ivan Niven's proof of the irrationality of pi is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of pi. This simplifying concept yields a more motivated proof of the irrationality of pi and pi squared.
[419]
vixra:1712.0353 [
pdf]
Goldbach Conjecture Proof
In this paper, we are going to give the proof of the Goldbach conjecture by introducing the lemma which implies Goldbach conjecture. first of all we are going to prove that the lemma implies Goldbach conjecture and in the following we are going to prove the validity of the lemma by using Chébotarev-Artin theorem's, Mertens formula and the Principle of inclusion - exclusion of Moivre
[420]
vixra:1712.0352 [
pdf]
Legendre Conjecture
In this paper, we are going to give the proof of legendre conjecture by using the Chebotarev -Artin 's theorem ,Dirichlet arithmetic theorem and the principle inclusion-exclusion of Moivre
[421]
vixra:1711.0291 [
pdf]
The Irrationality of Trigonometric and Hyperbolic Functions
This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.
[422]
vixra:1711.0258 [
pdf]
The Squared Case of Pi^n is Irrational Gives Pi is Transcendental
This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.
[423]
vixra:1711.0236 [
pdf]
Some Elementary Identities in Q-Series and the Generating Functions of the (M,k)-Capsids and (M, R1, R2)-Capsids
We demonstrate some elementary identities for q-series involving the q-Pochhammer symbol, as well as an identity involving the generating functions of the (m,k)-capsids and (m, r1, r2)-capsids.
[424]
vixra:1711.0202 [
pdf]
Sums of Arctangents and Sums of Products of Arctangents
We present new infinite arctangent sums and infinite sums of products of arctangents. Many previously known evaluations appear as special cases of the general results derived in this paper.
[425]
vixra:1711.0130 [
pdf]
The Irrationality and Transcendence of e Connected
Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.
[426]
vixra:1711.0127 [
pdf]
Demonstration de la Conjecture de Polignac
In this paperwe give the proof Polignac Conjecture by using Chebotarev -Artin theorem ,Mertens formula and Poincaré sieve For doing that we prove that .Let's X be an arbitrarily large real number and n an even integer we prove that there are many primes p such that p+n is prime between sqrt(X) and X
[427]
vixra:1710.0205 [
pdf]
An Approximation to the Prime Counting Function Through the Sum of Consecutive Prime Numbers
In this paper it is proved that the sum of consecutive prime numbers under the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, it is proved another asymptotic relationship between the sum of the first prime numbers up to the integer part of the square root of a given natural number and the prime counting function.
[428]
vixra:1710.0145 [
pdf]
Visualizing Zeta(n>1) and Proving Its Irrationality
A number system is developed to visualize the terms and partials of zeta(n>1). This number system consists of radii that generate sectors. The sectors have areas corresponing to all rational numbers and can be added via a tail to head vector addition. Dots on the circles give an un-ambiguous cross reference to decimal systems in all bases. We show, in the proof section of this paper, first that all partials require decimal bases greater than the last denominator used in the partial, then that this can be used to make a sequence of nested intervals with rational endpoints. Using Cantor's Nested Interval theorem this gives the convergence point of zeta series and disallows rational values, thus proving the irrationality of zeta(n>1).
[429]
vixra:1709.0408 [
pdf]
Fermat's Proof Of Fermat's Last Theorem
Employing only basic arithmetic and algebraic techniques that would have been known to Fermat, and utilizing alternate computation methods for arriving at $\sqrt[n]{c^n}$, we identify a governing relationship between $\sqrt{(a^2 + b^2)}$ and $\sqrt[n]{(a^n + b^n)}$ (for all $n > 2$), and are able to establish that $c = \sqrt[n]{(a^n + b^n)}$ can never be an integer for any value of $n > 2$.
[430]
vixra:1708.0380 [
pdf]
Division by Zero and the Arrival of Ada
Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.
[431]
vixra:1708.0255 [
pdf]
New Idea of the Goldbach Conjecture
A new idea of the Goldbach conjecture has been studied, it is that the even number is more bigger, the average form of the sum of two primes are more larger too. And then, we prove that every sufficiently large even number is the sum of two primes.
[432]
vixra:1707.0258 [
pdf]
A New Result About Prime Numbers: Lim N→+∞ N/(p(n) − N(ln N + ln ln N − 1)) = +∞
In this short paper we propose a new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞ .
[433]
vixra:1707.0176 [
pdf]
Uncertainty and the Lonely Runner Conjecture
By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
[434]
vixra:1707.0152 [
pdf]
A Conjecture About Prime Numbers Assuming the Riemann Hypothesis
In this paper we propose a conjecture about prime numbers. Based on the result of Pierre Dusart stating that the n th prime number is smaller than n(ln n + ln ln n − 0.9484) for n ≥ 39017 we propose that the n th prime number is smaller than n(ln n + ln ln n − 1+) when n → +∞.
[435]
vixra:1707.0086 [
pdf]
Zeta Function and Infinite Sums
I have come to the conclusion, after finishing a first reection on infinite sums, that all the functions which are written in the form of an infinite sum are written according to the famous Zeta function, this statement is explicitly presented in this article.
[436]
vixra:1707.0023 [
pdf]
An Interval Unifying Theorem About Primes
In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.
[437]
vixra:1707.0020 [
pdf]
One, Zero, Ada, and Cyclic Number System
Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.
[438]
vixra:1706.0408 [
pdf]
A New Sufficient Condition by Euler Function for Riemann Hypothesis
The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known Robin theorem, and prove that the NSC holds for all odd and some even numbers while, the NSC holds for any even numbers under a certain condition, which would be called the condition (d).
[439]
vixra:1706.0407 [
pdf]
An Upper Bound for Error Term of Mertens' Formula
In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.
[440]
vixra:1706.0288 [
pdf]
On Fermat's Last Theorem - An Elementary Approach
An attempt of using elementary approach to prove Fermat's last theorem (FLT) is given. For infinitely many prime numbers, Case I of the FLT can be proved using this approach. Furthermore, if a conjecture proposed in this paper is true (k-3 conjecture), then case I of the FLT is proved for all prime numbers. For case II of the FLT, a constraint for possible solutions is obtained.
[441]
vixra:1706.0112 [
pdf]
On the Quantum Differentiation of Smooth Real-Valued Functions
Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown. Keywords: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint
[442]
vixra:1706.0111 [
pdf]
On the Link Between Finite Differences and Derivatives of Polynomials
The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.
[443]
vixra:1705.0289 [
pdf]
Distribution of the Residues and Cycle Counting
In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions. We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.
[444]
vixra:1705.0142 [
pdf]
On the Riemann Hypothesis, Complex Scalings and Logarithmic Time Reversal
An approach to solving the Riemann Hypothesis is revisited within the framework of the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance. The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation $ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $. A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$, and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively. A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.
[445]
vixra:1705.0115 [
pdf]
A Criterion Arising from Explorations Pertaining to the Oesterle-Masser Conjecture
Using an extension of the idea of the radical of a number, as well as a few other ideas, it is indicated as to why one might expect the Oesterle-Masser conjecture to be true. Based on structural elements arising from this proof, a criterion is then developed and shown to be potentially sufficient to resolve two relatively deep conjectures about the structure of the prime numbers. A sketch is consequently provided as to how it might be possible to demonstrate this criterion, borrowing ideas from information theory and cybernetics.
[446]
vixra:1704.0196 [
pdf]
New Approximation Algorithms of Pi, Accelerated Convergence Formulas from N = 100 to N = 2m
We give algorithms for the calculation of pi. These algorithms can be easily developed in a linear manner and allows the calculation of pi with an infinite degree of convergence. Of course, the calculation of the second term passes through the first one, and it is necessary, as this type of algorithms, for a larger memory for calculations contrary to the formula BBP [1] whose execution corresponds to the order of the desired number. The advantage of our formulas in spite of the dificulty associated with extracting sin(x) lies in their degree of convergence, which is infinite, they prove the Borweins brothers hypothesis on the construction of algorithms At any speed as symbolized in our generic formula (8) of this paper. These formulas for the most part are totally new : We had found several other formulas of pi l
[447]
vixra:1704.0093 [
pdf]
A Finite Field Analogue for Appell Series F_{3}
In this paper we introduce a finite field analogue for the Appell series F_{3} and give some reduction formulae and certain generating functions for this function over finite fields.
[448]
vixra:1704.0029 [
pdf]
An Introduction to the N-Irreducible Sequents and the N-Irreducible Number
In this work, we introduce the $n$-irreducible sequents and the $n$-irreducible numbers defined with the help of the second order logic. We give many concrete examples of $n$-irreducible numbers and $n$-irreducible sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-irreducible iff the sequent is composed by some closed hypotheses and a $n$-irreducible formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-irreducible formula with $m>1$. The definition is motivated by the intuition that \Nathypo do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-irreducible sequent. Moreover, we postulate at second order of logic that \Nathypo are not chosen randomly: \Nathypo has the propriety to give the largest $n$-irreducible number $N_Z \NZ$ among a finite number of $n$-irreducible sequents. The Collatz conjecture, the Goldbach's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) $n$-irreducible axiom. Finally, two open questions remain: Can we prove that a natural number is not $n$-irreducible? If a $n$-irreducible number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-irreducible?
[449]
vixra:1703.0304 [
pdf]
Towards a Solution of the Riemann Hypothesis
In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) $s=\sigma+it$ of the zeta function, defined by: $$\zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$ have real part $\sigma= 1/2$. We give a proof that $\sigma= 1/2$ using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet $\eta$ function.
[450]
vixra:1703.0124 [
pdf]
Constant Quality of the Riemann Zeta's Non-Trivial Zeros
In this article we are closely examining Riemann zeta function's non-trivial zeros. Especially, we examine real part of non-trivial zeros. Real part of Riemann zeta function's non-trivial zeros is considered in the light of constant quality of such zeros. We propose and prove a theorem of this quality. We also uncover a definition phenomenons of zeta and Riemann xi functions. In conclusion and as an conclusion we observe Riemann hypothesis in perspective of our researches.
[451]
vixra:1703.0097 [
pdf]
High Degree Diophantine Equation C^q=a^p+b^p
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction the result that two finite numbers is with unequal logarithms in a corresponding module is proven, which result is applied to solving a kind of diophantine equation: $c^q=a^p+b^p$.
[452]
vixra:1703.0078 [
pdf]
Exponential Diophantine Equation
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction a result is obtained on two finite numbers with unequal logarithms, which result is applied to solving a kind of diophantine equations.
[453]
vixra:1703.0040 [
pdf]
The Asymptotic Riemann Hypothesis (ARH)
We propose in the present paper to consider the Riemann Hypothesis asympotically (ARH) ; it means when the imaginary part of the zero in the critical band is great. We show that the problem, expressed in these terms, is equivalent to the fact that an equation called the * equation has only a finite number of solutions, but we have not proved it.
[454]
vixra:1702.0265 [
pdf]
A New Simple Recursive Algorithm for Finding Prime Numbers Using Rosser's Theorem
In our previous work (The distribution of prime numbers: overview of n.ln(n), (1) and (2)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the previous is known. We use our method with five intervals and two values for n (see Methods and results). Our preliminary results show a reduced difference between the real next prime number and the number given by our algorithm. However long-term studies are required to better estimate the next prime number and to reduce the difference when n tends to infinity. Indeed an efficient algorithm is an algorithm that could be used in practical research to find new prime numbers for instance.
[455]
vixra:1702.0253 [
pdf]
The Distribution of Prime Numbers: Overview of N.ln(n)
The empirical formula giving the nth prime number p(n) is p(n) = n.ln(n) (from ROSSER (2)). Other studies have been performed (from DUSART for example (1)) in order to better estimate the nth prime number. Unfortunately these formulas don't work since there is a significant difference between the real nth prime number and the number given by the formulas. Here we propose a new model in which the difference is effectively reduced compared to the empirical formula. We discuss about the results and hypothesize that p(n) can be approximated with a constant defined in this work. As prime numbers are important to cryptography and other fields, a better knowledge of the distribution of prime numbers would be very useful. Further investigations are needed to understand the behavior of this constant and therefore to determine the nth prime number with a basic formula that could be used in both theoretical and practical research.
[456]
vixra:1701.0630 [
pdf]
Hyperspheres in Fermat's Last Theorem
This paper provides a potential pathway to a formal simple proof of Fermat's Last Theorem. The geometrical formulations of n-dimensional hypergeometrical models in relation to Fermat's Last Theorem are presented. By imposing geometrical constraints pertaining to the spatial allowance of these hypersphere configurations, it can be shown that a violation of the constraints confirms the theorem for n equal to infinity to be true.
[457]
vixra:1701.0618 [
pdf]
An Algorithmic Proof of the Twin Primes Conjecture and the Goldbach Conjecture
Abstract. This paper introduces proofs to several open problems in number theory, particularly the Goldbach Conjecture and the Twin Prime Conjecture. These two conjectures are proven by using a greedy elimination algorithm, and incorporating Mertens' third theorem and the twin prime constant. The argument is extended to Germain primes, Cousin Primes, and other prime related conjectures. A generalization is provided for all algorithms that result in an Euler product like\prod{\left(1-\frac{a}{p}\right)}.
[458]
vixra:1612.0278 [
pdf]
A Complete Proof of Beal Conjecture Followed by Numerical Examples
In 1997, Andrew Beal announced the following conjecture: \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or not a common factor. Three numerical examples are given.
[459]
vixra:1611.0390 [
pdf]
Proof of Bunyakovsky's Conjecture
Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate innitely many primes. The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences. As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and the binary Goldbach conjecture. The method is also used to prove that there are infinitely many primorial and factorial primes.
[460]
vixra:1611.0224 [
pdf]
Sieve of Collatz
The sieve of Collatz is a new algorithm to trace back the non-linear Collatz problem to a linear cross out algorithm. Until now it is unproved.
[461]
vixra:1611.0089 [
pdf]
The 3n ± p Conjecture: A Generalization of Collatz Conjecture
The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo Kakutani) and so on. Several various generalization of the Collatz conjecture has been carried. In this paper a new generalization of the Collatz conjecture called as the 3n ± p conjecture; where p is a prime is proposed. It functions on 3n + p and 3n - p, and for any starting number n, its sequence eventually enters a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is a special case of the 3n ± p conjecture when p is 1.
[462]
vixra:1610.0106 [
pdf]
A New 3n-1 Conjecture Akin to Collatz Conjecture
The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani's problem (after Shizuo Kakutani) and so on. In this paper a new conjecture called as the 3n-1 conjecture which is akin to the Collatz conjecture is proposed. It functions on 3n -1, for any starting number n, its sequence eventually reaches either 1, 5 or 17. The 3n-1 conjecture is compared with the Collatz conjecture.
[463]
vixra:1610.0065 [
pdf]
Lauricella Hypergeometric Series Over Finite Fields
In this paper we give a finite field analogue of the Lauricella hypergeometric series and obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. Some of these generalize some known results of Li \emph{et al} as well as several other well-known results.
[464]
vixra:1609.0384 [
pdf]
An Appell Series Over Finite Fields
In this paper we give a finite field analogue of one of the Appell series and obtain some transformation and reduction formulae and the generating functions for the Appell series over finite fields.
[465]
vixra:1609.0374 [
pdf]
Collatz Conjecture for $2^{100000}-1$ is True - Algorithms for Verifying Extremely Large Numbers
Collatz conjecture (or 3x+1 problem) is out for about 80 years. The verification of Collatz conjecture has reached to the number about 60bits until now. In this paper, we propose new algorithms that can verify whether the number that is about 100000bits (30000 digits) can return 1 after 3*x+1 and x/2 computations. This is the largest number that has been verified currently. The proposed algorithm changes numerical computation to bit computation, so that extremely large numbers (without upper bound) becomes possible to be verified. We discovered that $2^{100000}-1$ can return to 1 after 481603 times of 3*x+1 computation, and 863323 times of x/2 computation.
[466]
vixra:1609.0373 [
pdf]
Induction and Code for Collatz Conjecture or 3x+1 Problem
Collatz conjecture (or 3x+1 problem) has not been proved to be true or false for about 80 years. The exploration on this problem seems to ask for introducing a totally new method. In this paper, a mathematical induction method is proposed, whose proof can lead to the proof of the conjecture. According to the induction, a new representation (for dynamics) called ``code'' is introduced, to represent the occurred $3*x+1$ and $x/2$ computations during the process from starting number to the first transformed number that is less than the starting number. In a code $3*x+1$ is represented by 1 and $x/2$ is represented by 0. We find that code is a building block of the original dynamics from starting number to 1, and thus is more primitive for modeling quantitative properties. Some properties only exist in dynamics represented by code, but not in original dynamics. We discover and prove some inherent laws of code formally. Code as a whole is prefix-free, and has a unified form. Every code can be divided into code segments and each segment has a form $\{10\}^{p \geq 0}0^{q \geq 1}$. Besides, $p$ can be computed by judging whether $x \in[0]_2$, $x\in[1]_4$, or computed by $t=(x-3)/4$, without any concrete computation of $3*x+1$ or $x/2$. Especially, starting numbers in certain residue class have the same code, and their code has a short length. That is, $CODE(x \in [1]_4)=100,$ $CODE((x-3)/4 \in [0]_4)=101000,$ $CODE((x-3)/4 \in [2]_8)=10100100,$ $CODE((x-3)/4 \in [5]_8)=10101000,$ $CODE((x-3)/4 \in [1]_{32})=10101001000,$ $CODE((x-3)/4\in [3]_{32})=10101010000,$ $CODE((x-3)/4\in [14]_{32})=10100101000.$ The experiment results again confirm above discoveries. We also give a conjecture on $x \in [3]_4$ and an approach to the proof of Collatz conjecture. Those discoveries support the proposed induction and are helpful to the final proof of Collatz conjecture.
[467]
vixra:1609.0012 [
pdf]
On Transformation and Summation Formulas for Some Basic Hypergeometric Series
In this paper, we give an alternate and simple proofs for Sear’s three term 3 φ 2 transformation formula, Jackson’s 3 φ 2 transformation formula and for a nonterminating form of the q-Saalschütz sum by using q exponential operator techniques. We also give an alternate proof for a nonterminating form of the q-Vandermonde sum. We also obtain some interesting special cases of all the three identities, some of which are analogous to the identities stated by Ramanujan in his lost notebook.
[468]
vixra:1608.0449 [
pdf]
The Proof of Fermat's Last Theorem
We first prove a weak form of Fermat's Last Theorem; this unique lemma is key to the entire proof. A corollary and lemma follow inter-relating Pythagorean and Fermat solutions. Finally, we prove Fermat's Last Theorem.
[469]
vixra:1608.0439 [
pdf]
Cycle and the Collatz Conjecture
We study on the cycle in the Collatz conjecture and there is something surprise us. Our goal is to show that there is no Collatz cycle
[470]
vixra:1608.0429 [
pdf]
Expansion of the Euler Zigzag Numbers
This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.
[471]
vixra:1608.0144 [
pdf]
On the Properties of Generalized Multiplicative Coupled Fibonacci Sequence of R T H Order
Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r t h order with some new interesting properties.
[472]
vixra:1608.0140 [
pdf]
On the Properties of K Fibonacci and K Lucas Numbers
In this paper, some properties of k Fibonacci and k Lucas numbers are derived and proved by using matrices S and M. The identities we proved are not encountered in the k Fibonacci and k Lucasnumber literature.
[473]
vixra:1608.0135 [
pdf]
Determinantal Identities for K Lucas Sequence
In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence literature.
[474]
vixra:1608.0082 [
pdf]
Algorithm for Calculating Terms of a Number Sequence using an Auxiliary Sequence
A formula giving the $n$:th number of a sequence defined by a recursion formula plus initial value is deduced using generating functions. Of particular interest is the possibility to get an exact expression for the n:th term by means a recursion formula of the same type as the original one. As for the sequence itself it is of some interest that the original recursion is non-linear and the fact that the sequence grows very fast, the number of digits increasing more or less exponentially. Other sequences with the same rekursion span can be treated similarly.
[475]
vixra:1607.0437 [
pdf]
Infinite Arctangent Sums Involving Fibonacci and Lucas Numbers
Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here.
[476]
vixra:1607.0087 [
pdf]
Induction and Analogy in a Problem of Finite Sums
What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.
[477]
vixra:1606.0345 [
pdf]
A Polynomial Recursion for Prime Constellations
An algorithm for recursively generating the sequence of solutions of a prime constellation is described. The algorithm is based on an polynomial equation formed from the first n elements of the constellation. A root of this equation is the next element of the sequence.
[478]
vixra:1606.0109 [
pdf]
Using Periodic Functions to Determine Primes Composites and Factors
This paper discusses connections between periodic functions and primes, composites, and factors. Specifically, it shows how to use periodic functions to construct formulas for the following: the number of factors of a number, the specific factors of a number, the exact prime counting function and distribution, the nth prime, primes of any size, ”product polynomials” as periodic functions, primality and composite tests, prime gap finders, and ”anti-pulses.”
[479]
vixra:1606.0065 [
pdf]
The Real Parts of the Nontrivial Riemann Zeta Function Zeros
This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.
[480]
vixra:1604.0200 [
pdf]
The Density of Primes
The prime numbers has very irregular pattern. The problem of finding pattern in the prime numbers is the long-open problem in mathematics. In this paper, we try to solve the problem axiomatically. And we propose some natural properties of prime numbers.
[481]
vixra:1603.0362 [
pdf]
Various Arithmetic Functions and Their Applications
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, Romania). This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira - with his algorithmic thinking and knowledge of Mathcad.
[482]
vixra:1603.0254 [
pdf]
A Novel Approach to the Discovery of Ternary BBP-Type Formulas for Polylogarithm Constants
Using a clear and straightforward approach, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. A previously unproved degree~4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas.
[483]
vixra:1603.0227 [
pdf]
A New Binary BBP-type Formula for $\sqrt 5\,\log\phi$
Hitherto only a base 5 BBP-type formula is known for $\sqrt 5\log\phi$, where \mbox{$\phi=(\sqrt 5+1)/2$}, the golden ratio, ( i.e. Formula 83 of the April 2013 edition of Bailey's Compendium of \mbox{BBP-type} formulas). In this paper we derive a new binary BBP-type formula for this constant. The formula is obtained as a particular case of a BBP-type formula for a family of logarithms.
[484]
vixra:1603.0130 [
pdf]
La Conjecture de Beal : Une Démonstration Complète
In 1997, Andrew Beal announced the following conjecture : \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers. Three numerical examples are also given.
[485]
vixra:1601.0207 [
pdf]
Interpreting the Summation Notation When the Lower Limit is Greater Than the Upper Limit
In interpreting the sigma notation for finite summation, it is generally assumed that the lower limit of summation is less than or equal to the upper limit. This presumption has led to certain misconceptions, especially concerning what constitutes an empty sum. This paper addresses how to construe the sigma notation when the lower limit is greater than the upper limit
[486]
vixra:1512.0006 [
pdf]
An Asymptotic Robin Inequality
The conjectured Robin inequality for an integer $n>7!$ is $\sigma(n)<e^\gamma n \log \log n,$ where $\gamma$ denotes Euler constant, and $\sigma(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH). Writing $D(n)=e^\gamma n \log \log n-\sigma(n),$ and $d(n)=\frac{D(n)}{n},$ we prove unconditionally that $\liminf_{n \rightarrow \infty} d(n)=0.$ The main ingredients of the proof are an estimate for Chebyshev summatory function, and an effective version of Mertens third theorem due to Rosser and Schoenfeld. A new criterion for RH depending solely on $\liminf_{n \rightarrow \infty}D(n)$ is derived.
[487]
vixra:1511.0102 [
pdf]
On Generalized Harmonic Numbers, Tornheim Double Series and Linear Euler Sums
Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.
[488]
vixra:1510.0020 [
pdf]
A Complete Proof of Beal Conjecture-Final Version
In 1997, Andrew Beal announced the following conjecture: \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or not a common factor. Three numerical examples are given.
[489]
vixra:1509.0091 [
pdf]
Une Démonstration Élémentaire de la Conjecture de BEAL
En 1997, Andrew Beal avait annoncé la conjecture suivante: Soient A, B,C, m,n, et l des entiers positifs avec m,n,l >2. Si A^m + ^n = C^l alors A, B,et C ont un facteur commun. Nous commençons par construire le polynôme P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q avec p,q des entiers qui dépendent de A^m,B^n et C^l. Nous résolvons l'équation x^3-px+q=0 et nous obtenons les trois racines x_1,x_2,x_3 comme fonctions de p,q et d'un paramètre µ. Comme A^m,B^n,-C^l sont les seules racines de x^3-px+q=0, nous discutons les conditions pour que x_1,x_2,x_3 soient des entiers.
[490]
vixra:1508.0150 [
pdf]
Une Démonstration Elémentaire de la Conjecture de BEAL
En 1997, Andrew Beal \cite{B1} avait annoncé la conjecture suivante: \textit{Soient $A, B,C, m,n$, et $l$ des entiers positifs avec $m,n,l > 2$. Si $A^ m + B^n = C^l$ alors $A, B,$ et $C$ ont un facteur commun.} Nous commençons par construire le polynôme $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ avec $p,q$ des entiers qui dépendent de $A^m,B^n$ et $C^l$. Nous résolvons $x^3-px+q=0$ et nous obtenons les trois racines $x_1,x_2,x_3$ comme fonctions de $p,q$ et d'un paramètre $\theta$. Comme $A^m,B^n,-C^l$ sont les seules racines de $x^3-px+q=0$, nous discutons les conditions pourque $x_1,x_2,x_3$ soient des entiers. Des exemples numériques sont présentés.
[491]
vixra:1508.0005 [
pdf]
Two Conjectures in Number Theory
In this note, I propose a conjecture of generalization of the Lander, Parkin, and Selfridge conjecture; and a conjecture of generalization of the Beal’s conjecture.
[492]
vixra:1505.0203 [
pdf]
A Prospect Proof of the Goldbach's Conjecture
Based on, the well-ordering (N,<) of the set of natural numbers N, and some basic concepts of number theory, and using the proof by contradiction and the inductive proof on N, we prove that the validity of the Goldbach's statement: every even integer 2n > 4, with n > 2, is the sum of two primes. This result confirms the Goldbach conjecture, which allows to inserting it as theorem in number theory. Key Words: Well-ordering (N,<), basic concepts and theorems on number theory, the indirect and inductive proofs on natural numbers. AMS 2010: 11AXX, 11p32, 11B37.
[493]
vixra:1505.0077 [
pdf]
Set of All Pairs of Twin Prime Numbers is Infinite
In this paper we formulate an intuitive Hypothesis about a new aspect of a well known method called “Sieve of Eratosthenes” and then prove that set of natural numbers N = {1, 2, . . .} contains infinite number of pairs of twin primes.
[494]
vixra:1504.0239 [
pdf]
A Note About Power Function
In this paper described some new view and properties of the power function, the main aim of the work is to enter some new ideas. Also described expansion of power function, based on done research. Expansion has like Binominal theorem view, but algorithm not same.
[495]
vixra:1503.0058 [
pdf]
On the Natural Logarithm Function and its Applications
In present article, we create new integral representations for natural logarithm function, the Euler-Mascheroni constant, the natural logarithm of Riemann zeta function and the first derivative of Riemann zeta function.
[496]
vixra:1501.0201 [
pdf]
High Degree Diophantine Equation by Classical Number Theory
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By analysis in module and a careful constructing, a condition of non-solution of Diophantine Equation $a^p+b^p=c^q$ is proved that: $(a,b)=(b,c)=1,a,b>0,p,q>12$, $p$ is prime. The proof of this result is mainly in the last two sections.
[497]
vixra:1501.0129 [
pdf]
The Prime Number Formulas
Abstract There are many proposed partial prime number formulas, however, no formula can generate all prime numbers. Here we show three formulas which can obtain the entire prime numbers set from the positive integers, based on the Möbius function plus the “omega” function, or the Omega function, or the divisor function.
[498]
vixra:1410.0174 [
pdf]
Solving Diophantine Equations
In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation η(π(x)) = π(η(x)), where η is the Smarandache function and π is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful ”tools” for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.
[499]
vixra:1410.0066 [
pdf]
Notes on the Proof of Second Hardy-Littlewood Conjecture
In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture, namely that inequality $\pi(x)+\pi(y) > \pi (x+y)$ s examined, where $\pi(x)$ denotes the number of primes not exceeding $x$. It is shown that the inequality holds for all sufficiently large x and y. It has also been shown that for a given value of $y \geq 55$ the inequality $\pi(x)+\pi(y) > \pi (x+y)$ holds for all sufficiently large $x$. Finally, in the concluding section an argument has been given to completely settle the conjecture.
[500]
vixra:1409.0052 [
pdf]
When π(N) Does not Divide N
Let $\pi(n)$ denote the prime-counting function and let <br>% <br>$$f(n)=\left|\left\lfloor\log n-\lfloor\log n\rfloor-0.1\right\rfloor\right|\left\lfloor\frac{\left\lfloor n/\lfloor\log n-1\rfloor\right\rfloor\lfloor\log n-1\rfloor}{n}\right\rfloor\text{.}$$ <br>% <br>In this paper we prove that if $n$ is an integer $\ge 60184$ and $f(n)=0$, then $\pi(n)$ does not divide $n$. We also show that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $f(n)=1$. In addition, we prove that if $n\ge 60184$ and $n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\log n-1\rfloor$ located in the interval $[e^{\lfloor\log n-1\rfloor+1},e^{\lfloor\log n-1\rfloor+1.1}]$. This allows us to show that if $c$ is any fixed integer $\ge 12$, then in the interval $[e^c,e^{c+0.1}]$ there is always an integer $n$ such that $\pi(n)$ divides $n$. <p>Let $S$ denote the sequence of integers generated by the function $d(n)=n/\pi(n)$ (where $n\in\mathbb{Z}$ and $n>1$) and let $S_k$ denote the $k$th term of sequence $S$. Here we ask the question whether there are infinitely many positive integers $k$ such that $S_k=S_{k+1}$.
[501]
vixra:1409.0028 [
pdf]
The Proof for Non-existence of Perfect Cuboid
This paper shows the non-existence of perfect cuboid by using two tools, the first is representing Pythagoras triplets by two numbers and the second is realizing the impossibility of two similar equations for the same problem at the same time in different ways and the variables of one is relatively less than the other. When we express all Pythagoras triplets in perfect cuboid problem and rearrange it we can get a single equation that can express perfect cuboid. Unfortunately perfect cuboid has more than two similar equations that can express it and contradict one another.
[502]
vixra:1408.0195 [
pdf]
Comments on Recent Papers by S. Marshall Claiming Proofs of Several Conjectures in Number Theory
In a recent series of preprints S. Marshall claims to give proofs of several famous conjectures in number theory, among them the twin prime conjecture and Goldbach’s conjecture. A claimed proof of Beal’s conjecture would even imply an elemen- tary proof of Fermat’s Last Theorem. It is the purpose of this note to point out serious errors. It is the opinion of this author that it is safe to say that the claims of the above mentioned papers are lacking any basis.
[503]
vixra:1408.0003 [
pdf]
Co-Prime Gap N-Tuples that Sum to a Number and Other Algebraic Forms
We study the spacings of numbers co-prime to an even consecutive product of primes, P_m\# and its structure exposed by the fundamental theorem of prime sieving (FTPS). We extend this to prove some parts of the Hardy-Littlewood general prime density conjecture for all finite multiplicative groups modulo a primorial. We then use the FTPS to prove such groups have gap spacings which form arithmetic progressions as long as we wish. We also establish their densities and provide prescriptions to find them.
[504]
vixra:1407.0214 [
pdf]
A Fundamental Therorem Of Prime Sieving
We introduce a fundamental theorem of prime sieving (FTPS) and show how it illuminates structure on numbers co-prime to a random product of unique prime numbers. This theorem operates on the transition between the set of numbers co-prime to any product of unique prime numbers and the new set when another prime number is introduced in the product.
[505]
vixra:1407.0205 [
pdf]
An Application of Hardy-Littlewood Conjecture
In this paper, we assume that weaker Hardy-Littlewood Conjecture, we got a better upper bound of the exceptional real zero for a class of prime number module.
[506]
vixra:1406.0013 [
pdf]
Two Statements that Are Equivalent to a Conjecture Related to the Distribution of Prime Numbers
Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every $n\leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$. <p>Let $\pi[n+g(n),n+f(n)+g(n)]$ denote the amount of prime numbers in the interval $[n+g(n),n+f(n)+g(n)]$. Here we show that the conjecture described in [8] is equivalent to the statement that <br>% <br>$$\pi[n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$ <br>% <br>where <br>% <br>$$f(n)=\left(\frac{n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta}{|n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta|}\right)(1-\lfloor\sqrt{n}\rfloor)\text{, }g(n)=\left\lfloor1-\sqrt{n}+\lfloor\sqrt{n}\rfloor\right\rfloor\text{,}$$ <br>% <br>and $\beta$ is any real number such that $1<\beta<2$. We also prove that the conjecture in question is equivalent to the statement that <br>% <br>$$\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$ <br>% <br>where <br>% <br>$$S_n=n+\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor^2-\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor+1\text{.}$$ <br>% <br>We use this last result in order to create plots of $h(n)=\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]$ for many values of $n$.
[507]
vixra:1405.0023 [
pdf]
On a Simpler, Much More General and Truly Marvellous Proof of Fermat's Last Theorem (II)
English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem} which had for 358 years notoriously resisted all efforts to prove it. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the `truly marvellous' proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the `truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods which make use of Pythagoras theorem, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it.
[508]
vixra:1404.0098 [
pdf]
A Simple and General Proof of Beal's Conjecture
Using the same method that we used in the paper http://vixra.org/abs/1309.0154 to prove Fermat's Last Theorem in a simpler and truly marvellous way, we demonstrate that Beal's Conjecture yields -- in the simplest imaginable manner; to our effort to proving it.
[509]
vixra:1403.0942 [
pdf]
On Legendre’s Conjecture
Legendre’s conjecture, stated by Adrien-Marie Legendre ( 1752-1833 ), says there is always a prime between n2 and (n+1)2 . This conjecture is part of Landau’s problems. In this paper a proof of this conjecture is presented, using the method of generating prime numbers between consecutive squares, and proving that for every pair of consecutive squares with n >= 3 may be generated at least one prime number that belongs to the interval [n,(n+1)^2]
[510]
vixra:1312.0019 [
pdf]
The Cyclic Variation in the Density of Primes in the Intervals Defined by the Fibonacci Sequence
The Riemann R-function can be used to estimate the number of primes in an interval, where its accuracy is affected by the interval to which it is applied. Here, the successive intervals defined by the Fibonacci sequence will be shown to cause more cycles of R-function over- and under-estimation of primes than any of a large landscape of related sequences (calculations were continued up to one billion). The size of this landscape suggests that a special relationship exists between the Fibonacci sequence and the distribution of primes.
[511]
vixra:1311.0203 [
pdf]
Function Estimating Number of Pairs of Primes (P,q) for All Z of Form Z=p+q
This paper derives a function that estimates number of unique ways you can write z as z = p + q, where p and q are prime numbers, for every z E N that can be written in that form.
[512]
vixra:1310.0211 [
pdf]
Enumeration of All Primitive Pythagorean Triples with Hypotenuse Less Than or Equal to N
All primitive Pythagorean triples with hypotenuse less than or equal to N can be counted with the general formulas for generating sequences of Pythagorean triples ordered by $c-b$. The algorithm calculates the interval $(1,m)$ such that $c=N$ then $\nu$ from each $m$ is calculated to get the interval $(n_1,n_\nu)$ then $(m,n_\nu)=1$ is used for counting. It can be enumerated manually if $N$ is small but for large $N$ the algorithm must be implemented with any computer programming languages.
[513]
vixra:1310.0132 [
pdf]
On the Validity of the Riemann Hypothesis.
In this paper, we have established a connection between The Dirichlet series with the Mobius function $M (s) = \sum_{n=1}^{\infty} \mu (n) /n^s$ and a functional representation of the zeta function $\zeta (s)$ in terms of its partial Euler product. For this purpose, the Dirichlet series $M (s) $ has been modified and represented in terms of the partial Euler product by progressively eliminating the numbers that first have a prime factor 2, then 3, then 5, ..up to the prime number $p_r $ to obtain the series $M(s,p_r)$. It is shown that the series $M(s)$ and the new series $M(s,p_r)$ have the same region of convergence for every $p_r$. Unlike the partial sum of $M(s)$ that has irregular behavior, the partial sum of the new series exhibits regular behavior as $p_r$ approaches infinity. This has allowed the use of integration methods to compute the partial sum of the new series and to examine the validity of the Riemann Hypothesis.
[514]
vixra:1310.0044 [
pdf]
An Approximation for Primes
An approximation heuristic for the prime counting function Pi(x) is presented. It is numerically shown, that the heuristic is on average as good as Li(x)-0.5Li(sqrt(x)) for x up to 100,000.
[515]
vixra:1309.0154 [
pdf]
On a Simpler, Much More General and Truly Marvellous Proof of Fermat's Last Theorem (I)
English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat's Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time -- such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles's proof employs very advanced mathematical tools and methods that were not at all available in the known World during Fermat's days. Given that Fermat claimed to have had the `truly marvellous' proof, this fact that the proof only came after $358$ years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat's time, this has led many to doubt that Fermat actually did possess the `truly marvellous' proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat's Last Theorem actually yields to our efforts to prove it. This proof is so elementary that anyone with a modicum of mathematical prowess in Fermat's days and in the intervening 358 years could have discovered this very proof. This brings us to the tentative conclusion that Fermat might very well have had the `truly marvellous' proof which he claimed to have had and his `truly marvellous' proof may very well have made use of elementary arithmetic methods.
[516]
vixra:1308.0071 [
pdf]
Zeros Distribution of the Riemann Zeta-Function
Horizontal and vertical distributions of complex zeros of the Riemann zeta-function in the critical region are being found in general form in the paper on the basis of standard methods of function theory of complex variable.
[517]
vixra:1308.0026 [
pdf]
The Structurization of a Set of Positive Integers and Its Application to the Solution of the Twin Primes Problem
One of causes why Twin Primes problem was unsolved over a long period is that pairs of Twin Primes (PTP) are considered separately from other pairs of Twin Numbers (PTN). By purpose of this work is research of connections between different types of PTN. For realization of this purpose by author was developed the "Arithmetic of pairs of Twin Numbers" (APTN). In APTN are defined three types PTN. As shown in APTN all types PTN are connected with each other by relations which represent distribution of prime and composite positive integers less than n between them. On the basis of this relations (axioms APTN) are deduced formulas for computation of the number of PTN (NPTN) for each types. In APTN also is defined and computed Average value of the number of pairs are formed from odd prime and composite positive integers $ < n $ . Separately AVNPP for prime and AVNPC for composite positive integers. We also deduced formulas for computation of deviation NPTN from AVNPP and AVNPC. It was shown that if $n$ go to infinity then NPTN go to AVNPC or AVNPP respectively that permit to apply formulas for AVNPP and AVNPC to computation of NPTN. At the end is produced the proof of the Twin Primes problem with help of APTN. It is shown that if n go to infinity then NPTP go to infinity.
[518]
vixra:1307.0033 [
pdf]
Riemann's R-Function and the Distribution of Primes
Riemann's R-function is shown to alternately under- and over-estimate the number of primes in the intervals defined by the Fibonacci numbers, specifically from the interval [55,89] to the interval [317811,514229].
[519]
vixra:1304.0104 [
pdf]
Primes in the Intervals [kn,(k+1)n]
Abstract. In this paper, we prove: (a) for every integer n > 1 and a fixed integer k less than or equal to n, there exists a prime number p in between kn and (k + 1)n, and (b) conjectures of Legendre, Oppermann, Andrica, Brocard, and Improved version of Legendre conjecture as a particular case of (a).
[520]
vixra:1303.0088 [
pdf]
Foundations of Santilli Isonumber Theory
1.Foudations of Santilli isonumber theory.I:isonumber theory of the first kind;2.Santilli isonumber theory.II:isonumber theory of the second kind;3.Fermat last theorem and its applications;4.the proofs of binary Goldbach theorem using only partial primes;5.Santilli isocryptographic theory.Disproofs of Riemann hypothesis.
[521]
vixra:1303.0013 [
pdf]
Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes
The manuscript provides tables of abscissae and weights for Gauss-Laguerre integration on 64, 96 and 128 nodes, and abscissae and weights for Gauss-Hermite integration on 96 and 128 nodes.
[522]
vixra:1301.0021 [
pdf]
Number Systems Based On Logical Calculus
The reference \cite{Ref1} denote number systems with a logical calculus, but the form of natural numbers are not consistently in these number systems. So we rewrite number systems to correct the defect.
[523]
vixra:1211.0122 [
pdf]
On General Formulas for Generating Sequences of Pythagorean Triples Ordered by C-B
General formulas for generating sequences of Pythagorean triples ordered by c-b are studied in this paper. As computational proof, tables were made with a C++ script showing Pythagorean triples ordered by c-b and included as text files and screenshots. Furthermore, to enable readers to check and verify them, the C++ script which will interactively generate tables of Pythagorean triples from the computer console command line is attached. It can be run in Cling and ROOT CINT C/C++ interpreters or compiled.
[524]
vixra:1211.0116 [
pdf]
On the Interval $[n,9(n+3)/8]$
In this paper we prove that the interval $[n,9(n+3)/8]$ contains at least one prime number for every positive integer $n$. In order to achieve our goal, we use a result by Pierre Dusart and we also do manual calculations.
[525]
vixra:1211.0022 [
pdf]
On the Fibonacci Numbers, the Koide Formula, and the Distribution of Primes
The Koide formula from physics is modified for use with the reciprocals of primes found in the intervals defined by the Fibonacci numbers. This formula's resultant values are found to alternate lower, higher, lower, higher, etc. from the interval (5,8] to the interval (514229,832040]. This pattern, inverted, is also shown to occur when the corresponding results are computed for non-primes.
[526]
vixra:1209.0079 [
pdf]
Summability Calculus
In this manuscript, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite differences under a single simple umbrella. Using Summability Calculus, any given finite sum bounded by a variable n becomes immediately in analytic form. Not only can we differentiate and integrate with respect to the bound n without having to rely on an explicit analytic formula for the finite sum, but we can also deduce asymptotic expansions, accelerate convergence, assign natural values to divergent sums, and evaluate the finite sum for any complex value of n. This follows because the discrete definition of the simple finite sum embodies a unique natural continuation to the entire complex plane. Throughout the paper, many established results are strengthened such as the Bohr-Mollerup theorem, Stirling's approximation, Glaisher's approximation, and the Shannon-Nyquist sampling theorem. In addition, many celebrated theorems are extended and generalized such as the Euler- Maclaurin summation formula and Boole's summation formula. Finally, we show that countless identities that have been proved throughout the past 300 years by different mathematicians using different approaches can actually be derived in an elementary straightforward manner using the rules of Summability Calculus.
[527]
vixra:1208.0245 [
pdf]
The Arithmetic of Binary Representations of Even Positive Integer 2n and Its Application to the Solution of the Goldbach's Binary Problem
One of causes why Goldbach's binary problem was unsolved over a long period is that binary representations of even integer 2n (BR2n) in the view of a sum of two odd primes(VSTOP) are considered separately from other BR2n. By purpose of this work is research of connections between different types of BR2n. For realization of this purpose by author was developed the "Arithmetic of binary representations of even positive integer 2n" (ABR2n). In ABR2n are defined four types BR2n. As shown in ABR2n all types BR2n are connected with each other by relations which represent distribution of prime and composite positive integers less than 2n between them. On the basis of this relations (axioms ABR2n) are deduced formulas for computation of the number of BR2n (NBR2n) for each types. In ABR2n also is defined and computed Average value of the number of binary sums are formed from odd prime and composite positive integers $ < 2n $ (AVNBS). Separately AVNBS for prime and AVNBS for composite positive integers. We also deduced formulas for computation of deviation NBR2n from AVNBS. It was shown that if $n$ go to infinity then NBR2n go to AVNBS that permit to apply formulas for AVNBS to computation of NBR2n. At the end is produced the proof of the Goldbach's binary problem with help of ABR2n. For it apply method of a proof by contradiction in which we make an assumption that for any 2n not exist BR2n in the VSTOP then make computations at this conditions then we come to contradiction. Hence our assumption is false and forall $2n > 2$ exist BR2n in the VSTOP.
[528]
vixra:1208.0022 [
pdf]
On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures
Let $n\in\mathbb{Z}^+$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number? In this paper we show that this is actually the case for every $n \leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.
[529]
vixra:1207.0071 [
pdf]
Multiplication Modulo n Along The Primorials With Its Differences And Variations Applied To The Study Of The Distributions Of Prime Number Gaps. A.K.A. Introduction To The S Model
The sequence of sets of Z_n on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypothesis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers.
[530]
vixra:1205.0077 [
pdf]
(This Paper Has Been Withdrawn by the Author)
<em>This paper has been withdrawn by the author due to a flaw in the proof. / Este documento ha sido retirado por el autor debido a un error en la demostración.</em>
[531]
vixra:1205.0076 [
pdf]
A Finite Reflection Formula For A Polynomial Approximation To The Riemann Zeta Function
The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x)=⌊x−1⌋(x⌊x−1⌋+x−1) multiplied by s((s+1)/(s-1)). A finite-sum approximation to ζ(s) denoted by ζw(N;s) which has real roots at s=−1 and s=0 is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). A closed-form expression for the integral of ζw(N;s) over the interval s=-1..0 is given. The function χ(N;s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of ζw(N;s) and the reflection functions χ(N;s) are also provided. The values ζw(N;1−n) for integer values of n are found to be related to the Bernoulli numbers.
[532]
vixra:1203.0064 [
pdf]
The Goldbach Conjecture
The binary Goldbach conjecture asserts that every even integer greater than 4 is the sum of two primes. In order to prove this statement, we start by defining a kind of double sieve of Eratosthenes as follows. Given an even integer x, we sift out from [1, x] all those elements that are congruents to 0 modulo p, or congruents to x modulo p, where p is a prime less than the square root of x. So, any integer in the interval [sqrt{x}, x] that remains unsifted is a prime p for which either x-p = 1 or x-p is also a prime. Then, we introduce a new way to formulate this sieve, which we call the sequence of k-tuples of remainders. Using this tool, we obtain a lower bound for the number of elements in [1, x] that survives the sifting process. We prove, for every even number x greater than the square of 149, that there exist at least 3 integers in the interval [ 1, x ] that remains unsifted. This proves the binary Goldbach conjecture for every even number x greater than the square of 149, which is our main result.
[533]
vixra:1111.0038 [
pdf]
On a Strengthened Hardy-Hilbert�s Type Inequality
In this paper, by using the Euler-Maclaurin expansion for the zeta function and estimating the weight function effectively, we derive a strengthenment of a Hardy-Hilbert�s type inequality proved by W.Y. Zhong. As applications, some particular results are considered. work.
[534]
vixra:1107.0039 [
pdf]
Pseudo-Smarandache Functions of First and Second kind
In this paper we de ne two kinds of Pseudo-Smarandache functions. We have investigated more than fifty terms of each pseudoSmarandache function. We have proved some interesting results and properties of these functions.
[535]
vixra:1102.0058 [
pdf]
The Powers of π are Irrational
Transcendence of a number implies the irrationality of powers of a number, but in the case of π there are no separate proofs that powers of π are irrational. We investigate this curiosity. Transcendence proofs for e involve what we call Hermite�s technique; for π�s transcendence Lindemann�s adaptation of Hermite�s technique is used. Hermite�s technique is presented and its usage is demonstrated with irrationality proofs of e and powers of e. Applying Lindemann�s adaptation to a complex polynomial, π is shown to be irrational. This use of a complex polynomial generalizes and powers of π are shown to be irrational. The complex polynomials used involve roots of i and yield regular polygons in the complex plane. One can use graphs of these polygons to visualize various mechanisms used to proof π<sup>2</sup>, π<sup>3</sup>, and π<sup>4</sup> are irrational. The transcendence of π and e are easy generalizations from these irrational cases.
[536]
vixra:1101.0091 [
pdf]
Fermat Last Theorem And Riemann Hypothesis (6)
1637 Fermat wrote: �It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.� (6)
[537]
vixra:1101.0090 [
pdf]
Fermat Last Theorem And Riemann Hypothesis (5)
1637 Fermat wrote: �It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.� (5)
[538]
vixra:1101.0089 [
pdf]
Fermat Last Theorem And Riemann Hypothesis (4)
1637 Fermat wrote: �It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.� (4)
[539]
vixra:1011.0077 [
pdf]
On "Discovering and Proving that π is Irrational"
We discuss the logical fallacies in an article appeared in The American Mathematical Monthly [6], and present the historical origin and motivation of the simple proofs of the irrationality of π.
[540]
vixra:1010.0019 [
pdf]
Research on Number Theory and Smarandache Notions
This Book is devoted to the proceedings of the Sixth International Conference on Number Theory and Smarandache Notions held in Tianshui during April 24-25, 2010. The organizers were myself and Professor Wangsheng He from Tianshui Normal University. The conference was supported by Tianshui Normal University and there were more than 100 participants. We had one foreign guest, Professor K.Chakraborty from India. The conference was a great success and will give a strong impact on the development of number theory in general and Smarandache Notions in particular. We hope this will become a tradition in our country and will continue to grow. And indeed we are planning to organize the seventh conference in coming March which will be held in Weinan, a beautiful city of shaanxi.
[541]
vixra:1010.0006 [
pdf]
Power Structures in Finite Fields and the Riemann Hypothesis
Some tools are discussed, in order to build power structures of primitive roots in finite fields for any order q<sup>k</sup>; relations between distinct roots are deduced from m- and shift-and-add- sequences. Some heuristic computational techniques, where information in a m- sequence is built from below, are proposed. Full settlement is finally viewed in a physical scenario, where a path leading to the Riemann Hypothesis can be enlighted.
[542]
vixra:1008.0061 [
pdf]
Some Properties of the Pseudo-Smarandache Function
Charles Ashbacher [1] has posed a number of questions relating to the pseudo-Smarandache function Z(n). In this note we show that the ratio of consecutive values Z(n + 1)/Z(n) and Z(n - 1)/Z(n) are unbounded; that Z(2n)/Z(n) is unbounded; that n/Z(n) takes every integer value infinitely often; and that the series Σ<sub>n</sub> 1/Z(n)<sup>α</sup> is convergent for any α > 1.
[543]
vixra:1004.0028 [
pdf]
Disproofs of Riemann's Hypothesis
As it is well known, the Riemann hypothesis on the zeros of the ζ(s) function has been assumed to be true in various basic developments of the 20-th century mathematics, although it has never been proved to be correct. The need for a resolution of this open historical problem has been voiced by several distinguished mathematicians. By using preceding works, in this paper we present comprehensive disproofs of the Riemann hypothesis. Moreover, in 1994 the author discovered the arithmetic function J<sub>n</sub>(ω) that can replace Riemann's ζ(s) function in view of its proved features: if J<sub>n</sub>(ω) ≠ 0, then the function has infinitely many prime solutions; and if J<sub>n</sub>(ω) = 0, then the function has finitely many prime solutions. By using the Jiang J<sub>2</sub>(ω) function we prove the twin prime theorem, Goldbach's theorem and the prime theorem of the form x<sup>2</sup> + 1. Due to the importance of resolving the historical open nature of the Riemann hypothesis, comments by interested colleagues are here solicited.
[544]
vixra:1004.0027 [
pdf]
Foundations of Santilli's Isonumber Theory
In my works (see the bibliography at the end of the Preface) I often expressed the view that the protracted lack of resolution of fundamental problems in science signals the needs of basically new mathematics. This is the case, for example, for: quantitative representations of biological structures; resolution of the vexing problem of grand-unification; invariant treatment of irreversibility at the classical and operator levels; identification of hadronic constituents definable in our spacetime; achievement of a classical representation of antimatter; and other basic open problems.
[545]
vixra:1002.0024 [
pdf]
A Derivation of π(n) Based on a Stability Analysis of the Riemann-Zeta Function
The prime-number counting function π(n), which is significant in the prime number theorem, is derived by analyzing the region of convergence of the real-part of the Riemann-Zeta function using the unilateral z-transform. In order to satisfy the stability criteria of the z-transform, it is found that the real part of the Riemann-Zeta function must converge to the prime-counting function.
[546]
vixra:0910.0012 [
pdf]
A New Formula for the Sum of the Sixth Powers of Fibonacci Numbers
Sloane's On-Line Encyclopedia of Integer Sequences incorrectly states a lengthy formula for the sum of the sixth powers of the first n Fibonacci numbers. In this paper we prove a more succinct formulation. We also provide an analogue for the Lucas numbers. Finally, we prove a divisibility result for the sum of certain even powers of the first n Fibonacci numbers.
[547]
vixra:0909.0034 [
pdf]
On Strategies Towards the Riemann Hypothesis :Fractal Supersymmetric QM and a Trace Formula
The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s<sub>n</sub> = 1/2 + iλ<sub>n</sub>. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros ) by recurring to P a weighted superposition of Weierstrass functions ΣW(x,p,D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit completion relation ( "trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied this could be another test of the Riemann Hypothesis. In an appendix we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of CT -invariant Quantum Mechanics, because < Ψ<sub>s</sub> | CT | Ψ<sub>s</sub> > ≠ 0 where s are the complex eigenvalues of the scaling-like operators.
[548]
vixra:0908.0098 [
pdf]
The Riemann Hypothesis is a Consequence of CT-Invariant Quantum Mechanics
The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s<sub>n</sub> = 1/2 + iλ<sub>n</sub>. By constructing a continuous family of scaling-like operators involving the Gauss-Jacobi theta series and by invoking a novel CT-invariant Quantum Mechanics, involving a judicious charge conjugation C and time reversal T operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.
[549]
vixra:0908.0079 [
pdf]
On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity and Renormalization Group
Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert-Polya proposal to find an operator whose spectrum reproduces the ordinates ρ<sub>n</sub> (imaginary parts) of the zeta zeros in the critical line : s<sub>n</sub> = 1/2 + iρ<sub>n</sub> A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels E<sub>n</sub> = ρ<sub>n</sub>, when the boundary conditions Ψ<sub>E</sub> (x = -∞) = ± Ψ<sub>E</sub> (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation x = x(V) = π/2(dN(V)/dV), where the functional form of N(V) is given by the full-fledged Riemann-von Mangoldt counting function of the zeta zeros, including the <i>fluctuating</i> as well as the O(E<sup>-n</sup>) terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to <i>nonperiodic</i> crystal-like structures) in <i>integer</i> multiples of π follows from the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large x (O(1/logx)) has a one-to-one correspondence with the asymptotic limit of the <i>inverse</i> average density of the zeta zeros in the critical line suggesting intriguing connections to the Renormalization Group program.