[16] **viXra:1410.0174 [pdf]**
*submitted on 2014-10-27 11:29:15*

**Authors:** Octavian Cira, Florentin Smarandache

**Comments:** 252 Pages.

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.

**Category:** Number Theory

[15] **viXra:1410.0142 [pdf]**
*submitted on 2014-10-23 05:19:00*

**Authors:** Denise Vella-Chemla

**Comments:** 23 Pages.

We propose a modelization of binary Goldbach's decompositions in a 4 letters language that permits to envisage this problem in a new way.

**Category:** Number Theory

[14] **viXra:1410.0140 [pdf]**
*submitted on 2014-10-23 05:48:34*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I will define two interesting classes of odd composites often met (by the author of this paper) in the study of Fermat pseudoprimes, which might also have applications in the study of big semiprimes or in other fields. This two classes of composites n = p(1)*...*p(k), where p(1), ..., p(k) are the prime factors of n are defined in the following way: p(j) – p(i) + 1 is a prime or a power of a prime, respectively p(i) + p(j) – 1 is a prime or a power of prime for any p(i), p(j) prime factors of n such that p(1) ≤ p(i) < p(j) ≤ p(k).

**Category:** Number Theory

[13] **viXra:1410.0112 [pdf]**
*submitted on 2014-10-19 23:08:56*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Abstract: Twin prime conjecture states that there are infinite number of twin primes of the form p and p+2. Remarkable progress has recently been achieved by Y. Zhang to show that infinite primes that differ by large gap (~ 70 million) exist and this gap has been further narrowed to ~600 by others. We use an elementary approach to explore any obvious constraint that could limit the infinite nature of twin primes. Using Fermat’s little theorem as a surrogate for primality we derive an equation that suggests but not prove that twin primes can be infinite.

**Category:** Number Theory

[12] **viXra:1410.0107 [pdf]**
*submitted on 2014-10-19 06:16:32*

**Authors:** Zhang Tianshu

**Comments:** 16 Pages.

We first get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. Next expound relations between C and paf (ABC) by the symmetric law of odd numbers. Finally we have proven C ≤ Cε [paf (ABC)] 1+ ε such being the case A+B=C, and gcf (A, B, C) =1.

**Category:** Number Theory

[11] **viXra:1410.0068 [pdf]**
*submitted on 2014-10-13 07:58:09*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Every Integer stands in the center of two Integer-Primes

**Category:** Number Theory

[10] **viXra:1410.0066 [pdf]**
*replaced on 2014-10-18 07:04:05*

**Authors:** S. Roy

**Comments:** 5 Pages.

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.

**Category:** Number Theory

[9] **viXra:1410.0061 [pdf]**
*replaced on 2014-10-22 16:05:56*

**Authors:** Denise Vella-Chemla

**Comments:** 23 Pages.

On propose une modélisation des décompositions binaires de Goldbach dans un langage à 4 lettres qui permettent de découvrir des relations invariantes entre nombres de décompositions.

**Category:** Number Theory

[8] **viXra:1410.0059 [pdf]**
*submitted on 2014-10-12 09:18:30*

**Authors:** Chenglian LIU, Jian YE

**Comments:** 17 Pages.

The Goldbach's conjecture has plagued mathematicians for over two hundred and seventy years. Whether a professional or an amateur enthusiast, all
have been fascinated by this question. Why do mathematicians have no way to solve this problem? Up until now, Chen has been recognized for the most concise
proof his “1 + 2” theorem in 1973. In this article, the author will use elementary concepts to describe and indirectly prove the Goldbach conjecture.

**Category:** Number Theory

[7] **viXra:1410.0054 [pdf]**
*replaced on 2014-11-06 21:46:52*

**Authors:** Simon Plouffe

**Comments:** 6 Pages. replacement caused by a mismatch in versions

A presentation of various formulas is given. Many of these findings have no explanation
whatsoever. One is related to the mass ratio of the neutron and proton: 1.00137841917. Other
expression are given to the mass ratio of the neutron and the electron. They were found using a
variety of methods using either a HP‐15C calculator in 1988 to the current database of constants
of the author which consist of 13.155 billion entries.

**Category:** Number Theory

[6] **viXra:1410.0042 [pdf]**
*submitted on 2014-10-10 02:03:45*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make a conjecture which states that any Mersenne number (number of the form 2^n – 1, where n is natural) with odd exponent n, where n is greater than or equal to 3, also n is not a power of 3, is either prime either divisible by a 2-Poulet number. I also generalize this conjecture stating that any number of the form P = ((2^m)^n – 1)/3^k, where m is non-null positive integer, n is odd, greater than or equal to 5, also n is not a power of 3, and k is equal to 0 or is equal to the greatest positive integer such that P is integer, is either a prime either divisible by at least a 2-Poulet number (I will name this latter numbers Mersenne-Coman numbers) and I finally enunciate yet another related conjecture.

**Category:** Number Theory

[5] **viXra:1410.0041 [pdf]**
*submitted on 2014-10-10 02:04:05*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture which states that any Fermat number (number of the form 2^(2^n) + 1, where n is natural) is either prime either divisible by a 2-Poulet number. I also generalize this conjecture stating that any number of the form N = ((2^m)^p + 1)/3^k, where m is non-null positive integer, p is prime, greater than or equal to 7, and k is equal to 0 or is equal to the greatest positive integer such that N is integer, is either a prime either divisible by at least a 2-Poulet number (I will name this latter numbers Fermat-Coman numbers) and I finally enunciate yet another related conjecture.

**Category:** Number Theory

[4] **viXra:1410.0038 [pdf]**
*replaced on 2014-10-11 10:08:27*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k2^n-1 is introduced .

**Category:** Number Theory

[3] **viXra:1410.0025 [pdf]**
*submitted on 2014-10-05 16:25:44*

**Authors:** Md Zahid

**Comments:** 5 Pages.

2^e is rational or irrational number is not known. It is unsolved and open problem in analysis [1] .In this paper we proved that 2^e as irrational number. We attack the proof by method of contradiction. We assume that 2^e be rational number. Then we use some logarithms properties and simplification to get a relation between ‘e’ and assumed rational number, since we known that ‘e’ is irrational number, we use some further simplification and method to prove that 2^e is irrational number .

**Category:** Number Theory

[2] **viXra:1410.0017 [pdf]**
*submitted on 2014-10-04 07:44:59*

**Authors:** Zhang Tianshu

**Comments:** 22 Pages.

First we classify A, B and C according to their respective odevity, and ret rid of two kinds from AX+BY=CZ. Then affirm AX+BY=CZ such being the case A, B and C have a common prime factor by examples. After that, prove AX+BY≠CZ under these circumstances that A, B and C have not any common prime factor by mathematical analyses with the aid of the symmetric law of odd numbers. Finally we have proven that the Beal’s conjecture holds water after the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[1] **viXra:1410.0011 [pdf]**
*submitted on 2014-10-03 02:39:32*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

For every odd prime number exist a sum (x+y) so that (x-y) is also a prime number.
Every odd number is the difference of two square numbers
Every 4n number is the difference of two square numbers

**Category:** Number Theory