# Number Theory

## 1406 Submissions

 viXra:1406.0182 [pdf] submitted on 2014-06-30 01:00:00

### On Existence of Infinitely Many Primes of the Form x^2+1

Authors: Pingyuan Zhou
Comments: 5 Pages. Auther presents a conjecture related to distribution of a kind of special prime factors of Fermat numbers, which may imply existence of infinitely many primes of the form x^2+1.

It is well known that there are infinitely many prime factors of Fermat numbers, because prime factor of a Fermat prime is the Fermat prime itself but a composite Fermat number has at least two prime factors and Fermat numbers are pairwise relatively prime. Hence we conjecture that there is at least one prime factor (k^(1/2)*2^(a/2))^2+1 of Fermat number for F(n)-1≤a<F(n+1)-1 (n=0,1,2,3,…), where k^(1/2)is odd posotive integer, a is even positive integer and F(n) is Fermat number. The conjecture holds till a<F(4+1)-1=4294967296 from known evidences. Two corollaries of the conjecture imply existence of infinitely many primes of the form x^2+1, which is one of four basic problems about primes mentioned by Landau at ICM 1912.
Category: Number Theory

 viXra:1406.0181 [pdf] submitted on 2014-06-30 02:05:49

### On the Composite Terms in Sequence Generated from Mersenne-type Recurrence Relations

Authors: Pingyuan Zhou
Comments: 13 Pages. Author presents a conjecture on composite terms in so-called generilized Catalan-Mersenne number sequence, and tries to find a new way to imply existence of infinitely many composite Mersenne numbers whose exponets are primes.

We conjecture that there is at least one composite term in sequence generated from Mersenne-type recurrence relations. Hence we may expect that all terms are composite besides the first few continuous prime terms in Catalan-Mersenne number sequence and composite Mersenne numbers with exponets restricted to prime values are infinite.
Category: Number Theory

 viXra:1406.0161 [pdf] submitted on 2014-06-25 16:47:07

### Odd Pefect Number

Authors: Isaac Mor
Comments: 3 Pages. I got rid of the power of p when n=P*Q^2 with a simple proof

if n is an Odd Perfect Number then n=P*Q^2 I got rid of the power of P with a simple proof
Category: Number Theory

 viXra:1406.0155 [pdf] submitted on 2014-06-25 09:04:18

### Postulat de Bertrand

Authors: Arnaud Dhallewyn
Comments: 5 Pages. Tout droit réservé

Différente démonstration du postulat de Bertrand
Category: Number Theory

 viXra:1406.0147 [pdf] submitted on 2014-06-24 03:05:09

### Formula of the Solution of Diophantine Equations 2.

Authors: Andrey Loshinin

Collected back formulas of the solutions of certain Diophantine equations and their systems. These decisions were not known earlier.
Category: Number Theory

 viXra:1406.0142 [pdf] submitted on 2014-06-23 04:05:36

### Two Types of Pairs of Primes that Could be Associated to Poulet Numbers

Authors: Marius Coman

In this paper I combine two of my objects of study, the Poulet numbers and the different types of pairs of primes and I state two conjectures about few ways in which types of Poulet numbers could be associated with types of pairs of primes.
Category: Number Theory

 viXra:1406.0131 [pdf] replaced on 2014-07-02 17:51:03

### Proof of Beal’s Conjecture

Authors: Allan Cacdac

Using visualization of the pattern by providing examples and an elementary proof, we are able to prove and show that A,B and C will always have a common prime factor.
Category: Number Theory

 viXra:1406.0116 [pdf] submitted on 2014-06-18 11:23:35

Authors: Michael Pogorsky

Any odd perfect number is unknown. Simple analysis valid almost for all combinations of odd prime divisors proves that odd numbers constituted of them cannot be perfect.
Category: Number Theory

 viXra:1406.0114 [pdf] submitted on 2014-06-18 04:35:50

### Formula of the Solution of Diophantine Equations

Authors: Andrey Loshinin

Collected formula of the solution of Diophantine equations. All the formulas given to me. There are solutions of the equations in General form.
Category: Number Theory

 viXra:1406.0112 [pdf] submitted on 2014-06-18 05:16:00

### The best Formula of Prime Numbers

Authors: Xu Feng

The Best Formula on the Prime Numbers is awesome.
Category: Number Theory

 viXra:1406.0088 [pdf] submitted on 2014-06-14 11:50:07

### La Fonction Zêta de Riemann

Authors: Arnaud Dhallewyn
Comments: 102 Pages. Tout droit réservé

Présentation globale de la fonction zêta de Riemann
Category: Number Theory

 viXra:1406.0079 [pdf] submitted on 2014-06-13 15:03:54

### Few Possible Infinite Sets of Triplets of Primes Related in a Certain Way and an Open Problem

Authors: Marius Coman

In this paper I make three conjectures about a type of triplets of primes related in a certain way, i.e. the triplets of primes [p, q, r], where 2*p^2 – 1 = q*r and I raise an open problem about the primes of the form q = (2*p^2 – 1)/r, where p, r are also primes.
Category: Number Theory

 viXra:1406.0043 [pdf] submitted on 2014-06-08 03:12:08

### An Interesting Formula for Generating Primes and Five Conjectures About a Certain Type of Pairs of Primes

Authors: Marius Coman

In this paper I just enunciate a formula which often leads to primes and products of very few primes and I state five conjectures about the pairs of primes of the form [(q^2 - p^2 – 2*r)/2,(q^2 – p^2 + 2*r)/2], where p, q, r are odd primes.
Category: Number Theory

 viXra:1406.0030 [pdf] submitted on 2014-06-05 14:30:24

### Seven Conjectures on a Certain Way to Write Primes Including Two Generalizations of the Twin Primes Conjecture

Authors: Marius Coman

In this paper I make few conjectures about a way to write an odd prime p, id est p = q – r + 1, where q and r are also primes; two of these conjectures can be regarded as generalizations of the twin primes conjecture, which states that there exist an infinity of pairs of twin primes.
Category: Number Theory

 viXra:1406.0026 [pdf] replaced on 2015-02-22 06:52:47

### The proof of Goldbach's Conjecture

Authors: Ramón Ruiz
Comments: 34 Pages. This document has been written in Spanish. This research is based on an approach developed solely to demonstrate the binary Goldbach Conjecture and the Twin Primes Conjecture.

Goldbach's Conjecture statement: “Every even integer greater than 2 can be expressed as the sum of two primes”. Initially, to prove this conjecture, we can form two arithmetic sequences (A and B) different for each even number, with all the natural numbers that can be primes, that can added, in pairs, result in the corresponding even number. By analyzing the pairing process, in general, between all non-prime numbers of sequence A, with terms of sequence B, or vice versa, to obtain the even number, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula, to calculate the approximate number of pairs of primes that meet the conjecture for an even number x. The result of this formula is always equal or greater than 1, and it tends to infinite when x tends to infinite, which allow us to confirm that Goldbach's Conjecture is true. The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.
Category: Number Theory

 viXra:1406.0025 [pdf] replaced on 2015-02-22 13:56:43

### The proof of the Twin Primes Conjecture

Authors: Ramón Ruiz
Comments: 24 Pages. This document has been written in Spanish.

Twin Primes Conjecture statement: “There are infinitely many primes p such that (p + 2) is also prime”. Initially, to prove this conjecture, we can form two arithmetic sequences (A and B), with all the natural numbers, lesser than a number x, that can be primes and being each term of sequence B equal to its partner of sequence A plus 2. By analyzing the pairing process, in general, between all non-prime numbers of sequence A, with terms of sequence B, or vice versa, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula to calculate the approximate number of pairs of primes, p and (p + 2), that are lesser than x. The result of this formula tends to infinite when x tends to infinite, which allow us to confirm that the Twin Primes Conjecture is true. The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.
Category: Number Theory

 viXra:1406.0023 [pdf] replaced on 2014-06-08 18:05:07

### Analysis on the Critical Line

Authors: Ihsan Raja Muda Nasution

We analyze the anatomy of critical line. In this paper, we change the center of coordinate. Hence, we obtain the minimum quantity of the critical line. Meanwhile, we investigate further many characteristics of T and σ.
Category: Number Theory

 viXra:1406.0013 [pdf] submitted on 2014-06-02 21:07:58

### Two Statements that Are Equivalent to a Conjecture Related to the Distribution of Prime Numbers

Authors: Germán Paz
Comments: 16 Pages. 3 figures, Mathematica code; keywords: Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, Oppermann's conjecture, prime numbers, triangular numbers. This paper (with plots as ancillary files) is also available at arxiv.org/abs/1406.4801.

Let $n\in\mathbb{Z}^+$. In  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 $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.

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  is equivalent to the statement that
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$$\pi[n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$
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where
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$$f(n)=\left(\frac{n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta}{|n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta|}\right)(1-\lfloor\sqrt{n}\rfloor)\text{, }g(n)=\left\lfloor1-\sqrt{n}+\lfloor\sqrt{n}\rfloor\right\rfloor\text{,}$$
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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
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$$\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$
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where
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$$S_n=n+\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor^2-\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor+1\text{.}$$
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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 $n$.
Category: Number Theory