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

**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

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

**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

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

**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

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

**Authors:** Arnaud Dhallewyn

**Comments:** 5 Pages. Tout droit réservé

Différente démonstration du postulat de Bertrand

**Category:** Number Theory

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

**Authors:** Andrey Loshinin

**Comments:** 75 Pages.

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

**Category:** Number Theory

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

**Authors:** Marius Coman

**Comments:** 2 Pages.

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

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

**Authors:** Allan Cacdac

**Comments:** 4 Pages. Revised.

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

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

**Authors:** Michael Pogorsky

**Comments:** 2 pages

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

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

**Authors:** Andrey Loshinin

**Comments:** 46 Pages.

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

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

**Authors:** Xu Feng

**Comments:** 1 Page.

The Best Formula on the Prime Numbers is awesome.

**Category:** Number Theory

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

**Authors:** Arnaud Dhallewyn

**Comments:** 102 Pages. Tout droit réservé

Présentation globale de la fonction zêta de Riemann

**Category:** Number Theory

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

**Authors:** Marius Coman

**Comments:** 2 Pages.

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

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

**Authors:** Marius Coman

**Comments:** 2 Pages.

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

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

**Authors:** Marius Coman

**Comments:** 2 Pages.

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

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

**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

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

**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

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

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 8 Pages.

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

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

**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 [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 $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.

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

%

$$\pi[n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$

%

where

%

$$f(n)=\left(\frac{n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta}{|n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta|}\right)(1-\lfloor\sqrt{n}\rfloor)\text{, }g(n)=\left\lfloor1-\sqrt{n}+\lfloor\sqrt{n}\rfloor\right\rfloor\text{,}$$

%

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

%

$$\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$

%

where

%

$$S_n=n+\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor^2-\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor+1\text{.}$$

%

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

[1] **viXra:1406.0010 [pdf]**
*submitted on 2014-06-02 13:17:10*

**Authors:** Peter Schorer

**Comments:** 43 Pages.

We present several proofs of the 3x + 1 Conjecture, which asserts that repeated iterations of the function C(x) = (3x + 1)/(2^a) always terminate in 1 Here x is an odd, positive integer, and a is the largest positive integer such that the denominator divides the numerator. Our first proofs are based on a structure called “tuple-sets” that represents the 3x + 1 function in the “forward” (as opposed to the inverse) direction. All of our proofs are counter-intuitive, but not for that reason wrong. In “Most Recent Proof of the Conjecture” on page 11, we show that, because the “number” of tuples in each tuple-set is the same, regardless if counterexamples exist or not, and because the set of all non-counterexamples is the same, regardless if counterexampels exist or not, it follows that counterexamples do not exist. In our next two proofs, we show, by a simple inductive argument, that the contents of the set of all tuple-sets is the same, regardless if counterexamples exist or not, and from this we conclude that counterexamples do not exist. “Third Proof” is based on a structure called “recursive ‘spiral’s” that represents the 3x + 1 function in the inverse direction. We show that, because a large number of consecutive odd, positive integers are known, by computer test, to be non-counterexamples, it follows, by an inductive argument based on certain fundamental properties of recursive “spiral”s, that the set of all tuples in each infinite set of recursive “spiral”s is the same regardless if counterexamples exist or not. We infer from this that counterexamples do not exist.
As far as we have been able to determine, our approach to a solution of the Problem is original.

**Category:** Number Theory