[13] **viXra:1310.0250 [pdf]**
*submitted on 2013-10-28 21:47:16*

**Authors:** Jinhua Fei

**Comments:** 10 Pages.

This paper use the results of the value distribution theory , got a significant conclusion by Riemann hypothesis

**Category:** Number Theory

[12] **viXra:1310.0211 [pdf]**
*submitted on 2013-10-24 20:04:51*

**Authors:** Eduardo Calooy Roque

**Comments:** 4 Pages.

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.

**Category:** Number Theory

[11] **viXra:1310.0178 [pdf]**
*submitted on 2013-10-20 23:56:51*

**Authors:** Yibing Qiu

**Comments:** 1 Page.

talk about the twin prime conjecture in a new
perspective.

**Category:** Number Theory

[10] **viXra:1310.0177 [pdf]**
*submitted on 2013-10-20 09:58:38*

**Authors:** Sidharth Ghoshal

**Comments:** 7 Pages. This is Not a full paper but merely an idea that I wish to be document in the viXra repository

The following document outlines the concept of translating problems involving infinitely nested recursive arithmetic expression using a simple formal language.

**Category:** Number Theory

[9] **viXra:1310.0173 [pdf]**
*submitted on 2013-10-20 00:25:07*

**Authors:** Sidharth Ghoshal

**Comments:** 11 Pages.

This is an attempted proof of the twin prime conjecture. And though failed it offers good insight into a new method for attacking the proof by using asymptotically equivalent functions to the twin prime counting function.

**Category:** Number Theory

[8] **viXra:1310.0171 [pdf]**
*submitted on 2013-10-19 06:49:12*

**Authors:** Haji Talib Haydarli

**Comments:** 18 Pages.

In math there are some classic problems of number theory which have not been solved yet. Two of these problems are as below:
1. «Pair of twin primes» (where difference is equal to 2 such as pairs of twin prime numbers (3;5); (5;7); (11;13); …) are infinite.
2. «It is possible to show any even number, starting from 4, as a sum of two prime numbers »
2nd problem is known as «Goldbach-Euler problem».
In order to solve these problems we have compiled a table determining if the numbers like 6m-1 and 6m+1 are prime or composite. We have solved these problems as below by using some facts and conclusions besides compilede table.

**Category:** Number Theory

[7] **viXra:1310.0132 [pdf]**
*replaced on 2014-11-16 05:04:38*

**Authors:** Khalid Ibrahim

**Comments:** 43 Pages.

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.

**Category:** Number Theory

[6] **viXra:1310.0104 [pdf]**
*replaced on 2015-01-01 07:33:58*

**Authors:** Bertrand Wong

**Comments:** 15 Pages.

Euclid’s proof of the infinitude of the primes has generally been regarded as elegant. It is a proof by contradiction, or, reductio ad absurdum, and it relies on an algorithm which will always bring in larger and larger primes, an infinite number of them. However, the proof is also subtle and has been misinterpreted by some with one well-known
mathematician even remarking that the algorithm might not work for extremely large numbers. This paper, which is a revision/expansion of the author’s earlier paper published in an international mathematics journal in 2003, presents a strong argument which supports the validity of the twin primes conjecture, using reasoning similar to that of Euclid’s proof of the infinity of the primes.

**Category:** Number Theory

[5] **viXra:1310.0103 [pdf]**
*replaced on 2014-09-16 04:04:55*

**Authors:** Bertrand Wong

**Comments:** 8 Pages.

This paper, which is a revision/expansion of the author’s earlier paper published in an international mathematics journal in 2003, approaches the twin primes problem from a few different perspectives.

**Category:** Number Theory

[4] **viXra:1310.0102 [pdf]**
*replaced on 2014-12-30 09:30:46*

**Authors:** Bertrand Wong

**Comments:** 36 Pages.

This paper is a revision and expansion of two papers on the Goldbach conjecture which the author had published in an international mathematics journal in 2012. It presents insights on the conjecture gained over a period of many years.

**Category:** Number Theory

[3] **viXra:1310.0058 [pdf]**
*submitted on 2013-10-09 05:13:19*

**Authors:** Yibing Qiu

**Comments:** 2 Pages.

Abstract：With observations and speculation, this article puts forward a proposition
about twin primes that every pair of numbers of the form {2·6k ±1, k∈N} all be twin
primes. Proves the proposition statement is true applied Wilson’s theorem and
induction, show there are infinitely many twin primes of the form {2·6k ±1,k∈N},
and conclude the twin prime conjecture statement is true.

**Category:** Number Theory

[2] **viXra:1310.0048 [pdf]**
*replaced on 2016-09-15 13:20:35*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 10 Pages.

ABSTRACT: In this paper we present a method to get the prime counting function π(x) and
other arithmetical functions than can be generated by a Dirichlet series, first we use the general
variational method to derive the solution for a Fredholm Integral equation of first kind with
symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels
K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by
Dirichlet series, then we could find a solution for π(x) and ( ) ( )
n x
a n A x
≤
Σ = , solving δ J[φ ] = 0
for a given functional J, so the problem of finding a formula for the density of primes on the
interval [2,x], or the calculation of the coefficients for a given arithmetical function a(n), can be
viewed as some “Optimization” problems that can be attacked by either iterative or Numerical
methods (as an example we introduce Rayleigh-Ritz and Newton methods with a brief
description)
Also we have introduced some conjectures about the asymptotic behavior of the series
( ) n
n
p x
x p
≤
Ξ =Σ =Sn(x) for n>0 , and a new expression for the Prime counting function in terms
of the Non-trivial zeros of Riemann Zeta and its connection to Riemman Hypothesis and
operator theory.
• Keywords: =PNT (prime number

**Category:** Number Theory

[1] **viXra:1310.0044 [pdf]**
*replaced on 2013-10-09 03:54:22*

**Authors:** Martin Schlueter

**Comments:** 8 Pages.

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.

**Category:** Number Theory