[18] **viXra:1602.0359 [pdf]**
*submitted on 2016-02-28 07:19:45*

**Authors:** Ilija Barukčić

**Comments:** 5 pages. (C) Ilija Barukčić, Jever, Germany, 2016. All rights reserved.

The law of non-contradiction (LNC) is still one of the foremost among the principles of science and equally a fundamental principle of scientific inquiry too. Without the principle of non-contradiction we could not be able to distinguish between something true and something false. There are arguably many versions of the principle of non-contradiction which can be found in literature. The method of reductio ad absurdum itself is grounded on the validity of the principle of non-contradiction. To be consistent, a claim / a theorem / a proposition / a statement et cetera accepted as correct, cannot lead to a logical contradiction. In general, a claim / a theorem / a proposition / a statement et cetera which leads to the conclusion that +1 = +0 is refuted.

**Category:** Number Theory

[17] **viXra:1602.0346 [pdf]**
*submitted on 2016-02-27 09:04:31*

**Authors:** Zhang Tianshu

**Comments:** 28 Pages.

If every positive integer is able to be operated to 1 by the set operational rule of the Collatz conjecture, then begin with 1, we can get all positive integers after make infinitely many operations on the contrary of the set operational rule. In this article, we shall prove that the Collatz conjecture by the mathematical induction via the two-way operations is tenable.
Keywords: mathematical induction; the two-way operational rules; classify positive integers; the bunch of integers’ chains; operational routes

**Category:** Number Theory

[16] **viXra:1602.0343 [pdf]**
*submitted on 2016-02-27 06:48:52*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

Dirichlet’s theorem establishes that every arithmetic progression of the
form a+nb where gcd(a,b)=1 contains infinitely many primes for positive integers a
and b and n=1,2,3,4,………. . We show a simple proof for the existence of such an
arithmetic progression for every large prime. This also reveals a method to identify
arithmetic progressions on which a particular prime will appear.

**Category:** Number Theory

[15] **viXra:1602.0279 [pdf]**
*submitted on 2016-02-22 04:50:56*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: let P be a 3-Poulet number, d its least prime factor and q one of the other two prime factors; then the length of the period of the rational number P/d + d/P is for almost any P equal to q – 1 or equal to (q – 1)/n or equal to (q – 1)*n, where n positive integer.

**Category:** Number Theory

[14] **viXra:1602.0276 [pdf]**
*submitted on 2016-02-22 05:51:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: let P be a 2-Poulet number, d its least prime factor and q the other one; then the length of the period of the rational number P/d + d/P is equal to (q – 1)/n, where n positive integer.

**Category:** Number Theory

[13] **viXra:1602.0212 [pdf]**
*submitted on 2016-02-17 12:04:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that many numbers of the form 4*p^2 + 2*p + 1, where p and 2*p – 1 are odd primes, meet one of the following three conditions: (i) they are primes; (ii) they are equal to d*Q, where d is the least prime factor and Q the product of the others, and Q = (n*d – n + m)/m; (iii) they are equal to d*Q, where d is the least prime factor and Q the product of the others, and Q = (n*d + n - m)/m, and I make few related notes.

**Category:** Number Theory

[12] **viXra:1602.0205 [pdf]**
*submitted on 2016-02-17 05:11:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that many numbers of the form 4*p^2 – 2*p – 1, where p and 2*p – 1 are odd primes, meet one of the following three conditions: (i) they are primes; (ii) they are equal to d*Q, where d is the least prime factor and Q the product of the others, and Q = n*d – n + 1; (iii) they are equal to d*Q, where d is the least prime factor and Q the product of the others, and Q = n*d + n – 1, and I make few related notes.

**Category:** Number Theory

[11] **viXra:1602.0201 [pdf]**
*submitted on 2016-02-17 07:07:09*

**Authors:** Zhang Tianshu

**Comments:** 26 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds from AX+BY=CZ. Then, affirmed the existence of AX+BY=CZ in which case A, B and C have at least a common prime factor by certain of concrete equalities. After that, proved AX+BY≠CZ in which case A, B and C have not any common prime factor by the mathematical induction with the aid of the symmetric law of positive odd numbers after divide the inequality in four. Finally, reached a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[10] **viXra:1602.0200 [pdf]**
*submitted on 2016-02-17 07:15:37*

**Authors:** Zhang Tianshu

**Comments:** 29 Pages.

If every positive integer is able to be operated to 1 by the set operational rule of the Collatz conjecture, then begin with 1, we can get all positive integers after make infinitely many operations on the contrary of the set operational rule. In this article, we shall prove that the Collatz conjecture by the mathematical induction via the two-way operations is tenable.

**Category:** Number Theory

[9] **viXra:1602.0176 [pdf]**
*submitted on 2016-02-15 09:09:05*

**Authors:** Aaron Chau

**Comments:** 5 Pages.

Due to the Law 1: the numbers of the odds (subtractor) are more, and Law 2: the numbers of the odd integer (difference) are less; it is stressed in this article that the odd spaces which are located at the bottom line of the prime-odd pairs will never be filled in by the numbers of the odd integer, they have to be filled in by the primes (minuend) as well. Therefore, The Differences of More and Less in Number that Proves 1+1（P+2）.

**Category:** Number Theory

[8] **viXra:1602.0135 [pdf]**
*submitted on 2016-02-12 06:32:08*

**Authors:** Reuven Tint

**Comments:** 30 Pages. Original written Russian

Received and given the unique invariant identity on a set of arbitrary numerical systems,super concise proof of Fermat's Last Theorem, another version of the Beal Conjecture solution.

**Category:** Number Theory

[7] **viXra:1602.0133 [pdf]**
*submitted on 2016-02-11 21:56:29*

**Authors:** G.L.W.A Jayathilaka

**Comments:** 1 Page. This is the first real proof for beal conjecture.K can be there for any right angle triangle due to proportionality.

This is the proof of beal conjecture done by G.L.W.A Jayathilaka from Srilanka. See that K should be there for any right angle triangle due to proportionality.

**Category:** Number Theory

[6] **viXra:1602.0100 [pdf]**
*submitted on 2016-02-09 03:46:24*

**Authors:** Terubumi Honjou

**Comments:** 10 Pages.

Catalogue
Theoretical physics.
Chapter1. Current conditionsand issues.
Chapter 2 principle of particle oscillation
Chapter 3 principle of pulsating for dark energy
Chapter 4 4-dimensional space found
Chapter 5. Solve the mystery of the dark matter discovered
Chapter 6. Solve the mystery of the double slit experiment

**Category:** Number Theory

[5] **viXra:1602.0096 [pdf]**
*submitted on 2016-02-08 12:01:30*

**Authors:** Terubumi Honjou

**Comments:** 10 Pages.

Dark energy hypothesis proves the Riemann hypothesis.
[1]. And math's biggest challenge, prove the Riemann hypothesis.
[2]. Tackle the difficult Riemann hypothesis have been rejecting geniuses challenge for 150 years.
[3]. The biggest challenge Prime mystery, history of mathematics, Riemann proved challenging.
[4]. A new interpretation of the Riemann hypothesis. Zero point is all crosses the line.
[5]. Elementary pulsation principle opens the doors of Lehman expected certification.

**Category:** Number Theory

[4] **viXra:1602.0065 [pdf]**
*submitted on 2016-02-05 14:44:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In my previous paper “Bold conjecture on Fermat pseudoprimes” I stated that there exist a method to place almost any Fermat pseudoprime to base two (Poulet number) in an infinite subsequence of such numbers, defined by a quadratic polynomial, as a further term or as a starting term of such a sequence. In this paper I conjecture that there is yet another way to place a Poulet number in such a sequence defined by a polynomial, this time not necessarily quadratic.

**Category:** Number Theory

[3] **viXra:1602.0058 [pdf]**
*submitted on 2016-02-05 08:43:58*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In many of my previous papers I showed various methods, formulas and polynomials designed to generate sequences, possible infinite, of Poulet numbers or Carmichael numbers. In this paper I state that there exist a method to place almost any Fermat pseudoprime to base two (Poulet number) in such a sequence, as a further term or as a starting term.

**Category:** Number Theory

[2] **viXra:1602.0051 [pdf]**
*submitted on 2016-02-04 15:34:08*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In previous papers, I presented few applications of the multiples of the number 30 in the study of Carmichael numbers, i.e. in finding possible infinite sequences of such numbers; in this paper I shall list 15 probably infinite sequences of Poulet numbers that I discovered based on the multiples of the number 6.

**Category:** Number Theory

[1] **viXra:1602.0023 [pdf]**
*replaced on 2016-04-04 08:14:26*

**Authors:** Kolosov Petro

**Comments:** 10 Pages.

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems.

**Category:** Number Theory