# Number Theory

## 1904 Submissions

 viXra:1904.0592 [pdf] submitted on 2019-04-30 08:39:11

### Pi Formula

Authors: Edgar Valdebenito

In this note we recall a formula for Pi.
Category: Number Theory

 viXra:1904.0561 [pdf] replaced on 2019-10-15 03:14:29

### Proof that there Are no Odd Perfect Numbers

Authors: Kouji Takaki

We have obtained the conclusion that there are no odd perfect numbers.
Category: Number Theory

 viXra:1904.0517 [pdf] submitted on 2019-04-26 09:22:45

### A Simple Proof of the Legendre's Conjecture

Authors: Afmika, AF. Michael

This is a simple proof of the Legendre's conjecture. afmichael73@gmail.com afmichael.san@gmail.com
Category: Number Theory

 viXra:1904.0507 [pdf] replaced on 2019-05-02 07:08:03

### Remarks on Infinitesimal Amount of Riemann Zeta Zeros

Authors: Algirdas Antano Maknickas

This remarks prove, that Riemann zeta function has infinitesimal amount of zeros.
Category: Number Theory

 viXra:1904.0489 [pdf] submitted on 2019-04-26 01:20:05

### 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.
Category: Number Theory

 viXra:1904.0454 [pdf] submitted on 2019-04-23 08:38:55

### The Number Alpha=0.5*arccos(0.5*arccos(0.5*...))

Authors: Edgar Valdebenito

In this note we give some formulas related with the number: alpha=0.5*arccos(0.5*arccos(0.5*arccos(0.5*...))).
Category: Number Theory

 viXra:1904.0446 [pdf] submitted on 2019-04-23 18:28:40

### New Square Potentiation Algorithm

Authors: Zeolla Gabriel Martín

This document develops and demonstrates the discovery of a new square potentiation algorithm that works absolutely with all the numbers using the formula of the square of a binomial.
Category: Number Theory

 viXra:1904.0428 [pdf] replaced on 2019-10-30 16:53:51

### The Inconsistency of Arithmetic

Authors: Ralf Wüsthofen
Comments: 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, i.e. a set that is a structuring of the natural numbers starting from 3.
Category: Number Theory

 viXra:1904.0422 [pdf] replaced on 2019-05-02 04:00:00

Authors: Hajime Mashima

The three sides of the right triangle are rational numbers, and those with natural numbers are congruent numbers.
Category: Number Theory

 viXra:1904.0410 [pdf] submitted on 2019-04-21 15:17:55

### Fermat Equation for Hypercomplex Numbers

Authors: Yuly Shipilevsky

We consider generalized Fermat equation for hypercomplex numbers, in order to stimulate research and development of those generalization
Category: Number Theory

 viXra:1904.0386 [pdf] submitted on 2019-04-19 11:38:30

### Meaning of Irrational Numbers

In this short paper, I have tried to give a physical meaning to irrational numbers.
Category: Number Theory

 viXra:1904.0378 [pdf] submitted on 2019-04-19 21:33:45

### Riemann Zeta Function Nine Propositions

Authors: Pedro Caceres

The Riemann Zeta function or Euler–Riemann Zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series: () = ∑ ^(-z) from k=1,∞ The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of () in the critical line Re(z)=1/2. The Riemann Zeta function is one of the most studied and well known mathematical functions in history. In this paper, we will formulate nine new propositions to advance in the knowledge of the Riemann Zeta function
Category: Number Theory

 viXra:1904.0376 [pdf] replaced on 2019-05-14 02:42:38

### Nature Works the Way Number Works

Authors: Surajit Ghosh

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.
Category: Number Theory

 viXra:1904.0235 [pdf] submitted on 2019-04-12 17:45:46

### Riemann Hypothesis Yielding to Minor Effort

Authors: Arthur Shevenyonov

A set of distinct and elementary approaches, all embarking on the Euler-Riemann equivalence representing the zeta at zero, invariably point to a consistent solution structure. The Riemann Hypothesis as regards Re=1/2 gains full support as a core solution, albeit one amounting to a special nontrivial case warranting extensions and qualifications.
Category: Number Theory

 viXra:1904.0227 [pdf] replaced on 2019-05-26 08:48:53

### Algorithm Capable of Proving Goldbach’s Conjecture

Authors: Elizabeth Gatton-Robey

I created an algorithm capable of proving Goldbach’s Conjecture. This is not a claim to have proven the conjecture. The algorithm and all work contained in this document is original, so no outside sources have been used. This paper explains the algorithm then applies the algorithm with examples. The final section of the paper contains information to accompany my thoughts on why I believe Goldbach’s Conjecture can be proven with the use of my algorithm.
Category: Number Theory

 viXra:1904.0219 [pdf] submitted on 2019-04-11 18:49:36

### The Twin Power Conjecture

Authors: Yuly Shipilevsky

We consider a new conjecture regarding powers of integer numbers and more specifically, we are interesting in existence and finding pairs of integers: n ≥ 2 and m ≥ 2, such that n^m = m^n.
Category: Number Theory

 viXra:1904.0214 [pdf] replaced on 2019-05-27 18:47:01

### Solving Incompletely Predictable Problems Polignac's and Twin Prime Conjectures Using Information-Complexity Conservation

Authors: John Yuk Ching Ting
Comments: 18 Pages. Rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers are Incompletely Predictable numbers calculated using complex algorithm Sieve of Eratosthenes. Involving proposals that prime gaps and associated sets of prime numbers are infinite in magnitude, Twin prime conjecture deals with even prime gap 2 and is a subset of Polignac's conjecture which deals with all even prime gaps 2, 4, 6, 8, 10,.... Treated as Incompletely Predictable problems, we solve these conjectures with research method Information-Complexity conservation to get Plus Gap 2 Composite Number Continuous Law and Plus-Minus Gap 2 Composite Number Alternating Law.
Category: Number Theory

 viXra:1904.0146 [pdf] submitted on 2019-04-07 14:40:11

### A Tentative of The Proof of The ABC Conjecture - Case c=a+1

Comments: 10 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we consider the $abc$ conjecture in the case $c=a+1$. Firstly, we give the proof of the first conjecture that $c rad(ac)$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\ds \left(\frac{1}{\epsilon^2} \right)}$. A numerical example is presented.}
Category: Number Theory

 viXra:1904.0105 [pdf] submitted on 2019-04-06 00:57:16

### Discovery on Beal Conjecture

In this paper we give a proof for Beal's conjecture . Since the discovery of the proof of Fermat's last theorem by Andre Wiles, several questions arise on the correctness of Beal's conjecture. By using a very rigorous method we come to the proof. Let $\mathbb{G}=\{(x,y,z)\in \mathbb{N}^{3}: \min(x,y,z)\geq 3\}$ $\Omega_{n}=\{ p\in \mathbb{P}: p\mid n , p \nmid z^{y}-y^{z}\}$ , $$\mathbb{T}=\{(x,y,z)\in \mathbb{N}^{3}: x\geq 3,y\geq 3,z\geq 3\}$$ $\forall(x,y,z) \in \mathbb{T}$ consider the function $f_{x,y,z}$ be the function defined as : $$\begin{array}{ccccc} f_{x,y,z} & : \mathbb{N}^{3}& &\to & \mathbb{Z}\\ & & (X,Y,Z) & \mapsto & X^{x}+Y^{y}-Z^{z}\\ \end{array}$$ Denote by $$\mathbb{E}^{x,y,z}=\{(X,Y,Z)\in \mathbb{N}^{3}:f_{x,y,z}(X,Y,Z)=0\}$$ and $\mathbb{U}=\{(X,Y,Z)\in \mathbb{N}^{3}: \gcd(X,Y)\geq2,\gcd(X,Z)\geq2,\gcd(Y,Z)\geq2\}$ Let $x=\min(x,y,z)$ . The obtained result show that :if $A^{x}+B^{y}=C^{z}$ has a solution and $\Omega_{A}\not=\emptyset$, $\forall p \in \Omega_{A}$ , $$Q(B,C)=\sum_{j=1}^{x-1}[\binom{y}{j}B^{j}-\binom{z}{j}C^{j}]$$ has no solution in $(\frac{\mathbb{Z}}{p^{x}\mathbb{Z}})^{2}\setminus\{(\overline{0},\overline{0})\}$ Using this result we show that Beal's conjecture is true since $$\bigcup_{(x,y,z)\in\mathbb{T}}\mathbb{E}^{x,y,z}\cap \mathbb{U}\not=\emptyset$$ Then $\exists (\alpha,\beta,\gamma)\in \mathbb{N}^{3}$ such that $\min(\alpha,\beta,\gamma)\leq 2$ and $\mathbb{E}^{\alpha,\beta,\gamma}\cap \mathbb{U}=\emptyset$ The novel techniques use for the proof can be use to solve the variety of Diophantine equations . We provide also the solution to Beal's equation . Our proof can provide an algorithm to generate solution to Beal's equation
Category: Number Theory

 viXra:1904.0070 [pdf] replaced on 2019-10-07 09:24:41

### Elementary Proof of the Riemann Hypothesis

Authors: Stephen Marshall

The Riemann Hypothesis is one of the most important unresolved problems in Number Theory, it was first proposed by Bernhard Riemann, in 1859. For 160 years mathematicians have struggled with this problem to no avail. The difficulty of the Riemann Hypothesis is the main reason the hypothesis has remained unsolved. Although the Riemann Hypothesis remains unsolved, several mathematicians have proven other problems are the equivalent of the Riemann Hypothesis. In other words, if any of these equivalent criteria were solved, it would also solve the Riemann Hypothesis. Of particular interest to the author is a very elementary equivalent to the Riemann Hypothesis, Lagarias’s Elementary Version of the Riemann Hypothesis. In 2002, Jeffrey Lagarias proved that his problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. The beauty of the Lagarias’s Elementary Version of the Riemann Hypothesis, is that it is truly an elementary and very simple problem compared to the Riemann Hypothesis. The simplicity of Lagarias’s proof is what attracted the author to attempt to solve the Riemann Hypothesis. The author was very surprised at the simple proof he formulated using the elementary work of Lagarias. The moral of this story is that many times elementary or simple proofs exist to complex mathematical problems, this is one of those cases.
Category: Number Theory

 viXra:1904.0035 [pdf] submitted on 2019-04-02 14:51:02

### Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite

Authors: Stephen Marshall

Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... If n is a composite number then so is 2n − 1. More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89. Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. A new Mersenne prime was found in December 2017. As of January 2018, 50 Mersenne primes are now known. The largest known prime number 277,232,917 − 1 is a Mersenne prime. Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. Ever since M521 was proven prime in 1952, the largest known prime has always been Mersenne primes, which shows that Mersenne primes become large quickly. Since the prime numbers are infinite, and since all large primes discovered since 1952 have been Mersenne primes, this seems to be evidence indicating the infinitude of Mersenne primes since there has to continually be an infinite number of large primes, even if we don’t find them. Additional evidence, is that since prime numbers are infinite, there exist an infinite number of Mersenne numbers of form 2p – 1, meaning there exist an infinite number of Mersenne numbers that are candidates for Mersenne primes. However, as with 211 – 1, we know not all Mersenne numbers of form 2p – 1 are primes. All of this evidence makes it reasonable to conjecture that there exist an infinite number of Mersenne primes. First we will provide additional evidence indicating an infinite number of Mersenne primes. Then we will provide the proof.
Category: Number Theory

 viXra:1904.0034 [pdf] submitted on 2019-04-02 15:00:11

### Proof that Fermat Prime Numbers are Infinite

Authors: Stephen Marshall

Fermat prime is a prime number that are a special case, given by the binomial number of the form: Fn = 22n + 1, for n ≥ 0 They are named after Pierre de Fermat, a Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory. The only known Fermat primes are: F0 = 3 F1 = 5 F2 = 17 F3 = 257 F4 = 65,537 It has been conjectured that there are only a finite number of Fermat primes, however, we will use the same technique the author used to prove that the Mersenne primes are infinite, to prove the Fermat primes are infinite.
Category: Number Theory

 viXra:1904.0033 [pdf] submitted on 2019-04-02 15:07:04

### Proof that Wagstaff Prime Numbers are Infinite

Authors: Stephen Marshall

The Wagstaff prime is a prime number q of the form: q = (2^p- 1)/3 where, p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography. The New Mersenne conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third: 1. p = 2k ± 1 or p = 4k ± 3 for some natural number k. 2. 2p − 1 is prime (a Mersenne prime). 3. (2p + 1) / 3 is prime (a Wagstaff prime). There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving which is very time consuming. A Wagstaff prime can also be interpreted as a repunit prime of base , as if p is odd, as it must be for the above number to be prime. The first three Wagstaff primes are 3, 11, and 43 because The first few Wagstaff primes are: 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, … (sequence A000979 in the OEIS) As of October 2014, known exponents which produce Wagstaff primes or probable primes are: 3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, (all known Wagstaff primes) 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531 (Wagstaff probable primes) (sequence A000978 in the OEIS) In February 2010, Tony Reix discovered the Wagstaff probable prime: which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date. In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes: and, Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes. Note that when p is a Wagstaff prime, need not to be prime, the first counterexample is p = 683, and it is conjectured that if p is a Wagstaff prime and p>43, then is composite.
Category: Number Theory

 viXra:1904.0032 [pdf] submitted on 2019-04-02 15:13:48

### Proof of Landau’s Fourth Problem

Authors: Stephen Marshall