[18] **viXra:1412.0273 [pdf]**
*submitted on 2014-12-30 14:23:47*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Collatz conjecture states that starting with any positive integer n, we divide it by 2 if it is even or multiply it by 3 and add 1 if it is odd and repeat this algorithm on the answer always using the same odd or even rule, we will ultimately end up with an answer of 1. Here we prove this conjecture for a special integer which is the product of any prime number “p” greater than three with another positive odd integer “x” that has been derived by using the Fermat’s little theorem and is therefore unique for each prime. Thus we prove Collatz’s conjecture for a small fraction of positive integers “px” which would be expected to roughly represent the same proportion of integers as prime numbers.

**Category:** Number Theory

[17] **viXra:1412.0250 [pdf]**
*submitted on 2014-12-27 04:44:46*

**Authors:** Nicolae Bratu, Adina Cretan

**Comments:** 4 Pages.

The present paper is a fragment revised from the work [3], published only in Romanian. Using a new function, “cubic combination”, we can solve different problems. The novelty of this work consists in the deduction of an infinite number of third degree Ramanujan identities.

**Category:** Number Theory

[16] **viXra:1412.0248 [pdf]**
*submitted on 2014-12-27 05:22:19*

**Authors:** Nicolae Bratu

**Comments:** 8 Pages.

In the work “Disquisitiones Diophanticae”, published in 2006 in Romanian, I had gathered succinctly and schematized the content of the “Memorandum to the Romanian Academy” in 1983, concerning the Fermat’s Last Theorem. This paper demonstrates a lemma representing a completion of the algebraic method proposed by us to prove the Fermat’s Theorem.

**Category:** Number Theory

[15] **viXra:1412.0246 [pdf]**
*submitted on 2014-12-26 16:11:05*

**Authors:** Nicolae BRATU, Adina CRETAN

**Comments:** 6 Pages.

This paper has been updated and completed thanks to suggestions and critics coming from Dr. Mike Hirschhorn, from the University of New South Walles. We want to express our highest gratitude.
The paper appeared in an abbreviated form [6]. The present work is a complete form.
For the homogeneous diophantine equations:x2 + by2 + cz2 = w2 there are solutions in the literature only for particular values of the parameters b and c. These solutions were found by Euler, Carmichael, Mordell. They proposed a particular solution for this equation in [3]. This paper presents the general solution of this equation as functions of the rational parameters b, c and their divisors. As a consequence, we obtain the theorem that every positive integer can be represented as the sum of three squares, with at most one of them duplicated, which improves on the Fermat –Lagrange theorem

**Category:** Number Theory

[14] **viXra:1412.0236 [pdf]**
*submitted on 2014-12-25 04:38:19*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time compositeness test for numbers of the form 2*3^n-1 is introduced .

**Category:** Number Theory

[13] **viXra:1412.0228 [pdf]**
*submitted on 2014-12-24 01:42:47*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make an observation about an interesting formula based on the lesser prime p from a pair of twin primes, id est N = p^3 + 3*p^2 + 4*p + 1, that conducts sometimes to the result N = q*r, where q, r are primes such that r – q + 1 = p and sometimes to the result N = q*r, where at least one from q, r or both are composites such that r – q + 1 = p.

**Category:** Number Theory

[12] **viXra:1412.0223 [pdf]**
*replaced on 2015-01-02 23:09:53*

**Authors:** Martin Schlueter

**Comments:** 1 Page. This document is licensed under a Creative Commons (CC BY-NC-ND)

An (assumed) new relationship between the harmonic series $H_{n}$ and the natural logarithm $log(n)$ is presented.

**Category:** Number Theory

[11] **viXra:1412.0178 [pdf]**
*submitted on 2014-12-15 05:30:02*

**Authors:** Grzegorz Ileczko

**Comments:** 13 Pages.

The Riemann hypothesis is not proved by more, than 150 years. At this paper, I presented new solution for this problem. I found new trigonometrical form of Riemann's zeta function for negative numbers (n). This new form of zeta gives opportunity to prove the Riemann hypothesis. Presented proof isn’t complicated for trigonometrical form of zeta function.

**Category:** Number Theory

[10] **viXra:1412.0164 [pdf]**
*submitted on 2014-12-11 15:54:12*

**Authors:** Stephen Marshall

**Comments:** 5 Pages.

This paper presents a complete rebuttal of the paper Vixra 1408.0195v2 posted by Matthias Lesch on 13 September 2014. This rebuttal is in response to Vixra 1408.0195v2 where Matthias Lesch erroneously attempted to disprove six papers I published proving several conjectures in Number Theory. Specifically, these were papers Vixra:1408.0169, 1408.0174, 1408.0201, 1408.0209, and 1408.0212. This rebuttal paper is presented in the same format as Vixra 1408.0195v2 with necessary quotes from paper Vixra 1408.0195v2 to clarify rebuttals.

**Category:** Number Theory

[9] **viXra:1412.0150 [pdf]**
*replaced on 2015-04-08 05:00:13*

**Authors:** Marius Coman

**Comments:** 62 Pages.

In this book I define a function which allows the reduction to any non-null positive integer to one of the digits 1, 2, 3, 4, 5, 6, 7, 8 or 9. The utility of this enterprise is well-known in arithmetic; the function defined here differs apparently insignificant but perhaps essentially from the function modulo 9 in that is not defined on 0, also can’t have the value 0; essentially, the mar reduced form of a non-null positive integer is the digital root of this number but with the important distinction that is defined as a function such it can be easily used in various applications (divizibility problems, Diophantine equations), a function defined only on the operations of addition and multiplication not on the operations of subtraction and division. Some of the results obtained with this tool are a proof of Fermat’s last Theorem, cases n = 3 and n = 4, using just integers, no complex numbers and a Diophantine analysis of perfect numbers.
Note: I understand, in this book, the numbers denoted by “abc” as the numbers where a, b, c are digits, and the numbers denoted by “a*b*c” as the products of the numbers a, b, c.

**Category:** Number Theory

[8] **viXra:1412.0136 [pdf]**
*submitted on 2014-12-07 03:30:07*

**Authors:** Pingyuan Zhou

**Comments:** 16 Pages. This paper has been submitted to mathematical journal.

In this paper we give a proof of the strong Goldbach conjecture by studying limit status of original continuous Goldbach natural number sequence generated by original continuous odd prime number sequence. It implies the weak Goldbach conjecture. If a prime p is defined as Goldbach prime when GNL = p then Goldbach prime is the higher member of a twin prime pair, from which we will give a proof of the twin prime conjecture.

**Category:** Number Theory

[7] **viXra:1412.0124 [pdf]**
*submitted on 2014-12-06 03:01:39*

**Authors:** Barar Stelian Liviu

**Comments:** 20 Pages.

This paper is protected by copiryght from
16.07.2012
I want to present the paper tu the viXra .

**Category:** Number Theory

[6] **viXra:1412.0046 [pdf]**
*submitted on 2014-12-03 04:34:06*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures about two types of possible infinite sequences of primes obtained starting from any given prime which is the lesser term from a pair of twin primes for a possible infinite of positive integers which are not of the form 3*k – 1 respectively starting from any given positive integer which is not of the form 3*k - 1 for a possible infinite of lesser terms from pairs of twin primes.

**Category:** Number Theory

[5] **viXra:1412.0044 [pdf]**
*submitted on 2014-12-02 22:18:18*

**Authors:** Zhang Tianshu

**Comments:** 26 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 concrete examples. After that, prove AX+BY≠CZ such being the case A, B and C have not any common prime factor by the mathematical induction with the aid of the symmetric law of odd numbers after the decomposition of the inequality. Finally, reached such a conclusion that the Beal’s conjecture can hold water after the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[4] **viXra:1412.0042 [pdf]**
*submitted on 2014-12-03 02:29:23*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make eight conjectures about a type of numbers which I defined in a previous paper, “The notion of chameleonic numbers, a set of composites that «hide» in their inner structure an easy way to obtain primes”, in the following way: the non-null positive composite squarefree integer C not divisible by 2, 3 or 5 is such a number if the absolute value of the number P – d + 1 is always a prime or a power of a prime, where d is one of the prime factors of C and P is the product of all prime factors of C but d.

**Category:** Number Theory

[3] **viXra:1412.0041 [pdf]**
*replaced on 2014-12-03 08:13:58*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k*2^n-1 is introduced .

**Category:** Number Theory

[2] **viXra:1412.0039 [pdf]**
*submitted on 2014-12-02 12:33:58*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make four conjectures about a certain type of semiprimes which I defined in a previous paper, “Two exciting classes of odd composites defined by a relation between their prime factors”, in the following way: Coman semiprimes of the first kind are the semiprimes p*q with the property that q1 – p1 + 1 = p2*q2, where the semiprime p2*q2 has also the property that q2 – p2 + 1 = p3*q3, also a semiprime, and the operation is iterate until eventually pk – qk + 1 is a prime. I also defined Coman semiprimes of the second kind the semiprimes p*q with the property that q1 + p1 - 1 = p2*q2, where the semiprime p2*q2 has also the property that q2 + p2 - 1 = p3*q3, also a semiprime, and the operation is iterate until eventually pk + qk - 1 is a prime.

**Category:** Number Theory

[1] **viXra:1412.0036 [pdf]**
*submitted on 2014-12-02 10:12:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

There exist few distinct generalizations of Fermat numbers, like for instance numbers of the form F(k) = a^(2^k) + 1, where a > 2, or F(k) = a^(2^k) + b^(2^k) or Smarandache generalized Fermat numbers, which are the numbers of the form F(k) = a^(b^k) + c, where a, b are integers greater than or equal to 2 and c is integer such that (a, c) = 1. In this paper I observe two formulas based on a new type of generalized Fermat numbers, which are the numbers of the form F(k) = (a^(b^k) ± c)/d, where a, b are integers greater than or equal to 2 and c, d are positive non-null integers such that F(k) is integer.

**Category:** Number Theory