[26] **viXra:1512.0484 [pdf]**
*submitted on 2015-12-30 05:18:01*

**Authors:** Marius Coman

**Comments:** 7 Pages.

In this paper I present a simple list of polynomials (in one or two variables) and formulas having the property that they generate Carmichael numbers or Poulet numbers, polynomials and formulas that I have discovered over time.

**Category:** Number Theory

[25] **viXra:1512.0473 [pdf]**
*submitted on 2015-12-29 09:48:38*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make two conjectures on Super-Poulet numbers with two, respectively three prime factors.

**Category:** Number Theory

[24] **viXra:1512.0471 [pdf]**
*submitted on 2015-12-29 10:39:11*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper we conjecture that there exist an infinity of Poulet numbers of the form m*n – n + 1, where m is of the form 270*k + 13. Incidentally, verifying this conjecture, we found results that encouraged us to issue yet another conjecture, i.e. that there exist an infinity of numbers s of the form 270*k + 13 which are semiprimes s = p*q having the property that q – p + 1 is prime or power of prime.

**Category:** Number Theory

[23] **viXra:1512.0470 [pdf]**
*submitted on 2015-12-29 10:41:51*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper we conjecture that there exist an infinity of primes, respectively squares of primes, respectively semiprimes with a certain property, respectively Poulet numbers of the form (p + 270)*n – n + 1, for any p prime greater than or equal to 7, if p + 270 is also a prime number.

**Category:** Number Theory

[22] **viXra:1512.0468 [pdf]**
*submitted on 2015-12-29 09:08:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper we conjecture that the square of any prime greater than or equal to 5 can be written in one of the following three ways: (i) p*q + q – p; (ii) p*q*r + p*q – r; (iii) p*q*r + p – q*r, where p, q and r are odd primes. Incidentally, verifying this conjecture, we found results that encouraged us to issue yet another conjecture, i.e. that the square of any prime of the form 11 + 30*k can be written as 3*p*q + p – 3*q, where p and q are odd primes.

**Category:** Number Theory

[21] **viXra:1512.0467 [pdf]**
*submitted on 2015-12-29 09:09:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper we conjecture that any Carmichael number C can be written as C = (p + 270)*(n + 1) – n, where n non-null positive integer and p prime. Incidentally, verifying this conjecture, we found results that encouraged us to issue yet another conjecture, i.e. that there exist an infinity of Poulet numbers P2 that could be written as (P1 + n)/(n + 1) – 270, where n is non-null positive integer and P1 is also a Poulet number.

**Category:** Number Theory

[20] **viXra:1512.0428 [pdf]**
*submitted on 2015-12-25 21:31:37*

**Authors:** Chunxuan Jiang

**Comments:** 8 Pages.

On Oct.25,1991 without using any number theory we have proved Fermat last theorem

**Category:** Number Theory

[19] **viXra:1512.0394 [pdf]**
*replaced on 2016-07-13 19:50:32*

**Authors:** Hajime Mashima

**Comments:** 10 Pages.

"It multiplies the 1/2 when the number of nature is even. It is multiplied by 3 in the case of an odd number. Furthermore it adds 1. By repeating this calculation, it leads always to 1." This is referred to as the Collatz problem.

**Category:** Number Theory

[18] **viXra:1512.0391 [pdf]**
*submitted on 2015-12-21 16:03:55*

**Authors:** Kolosov Petro

**Comments:** 6 Pages.

This paper presents the way to make expansion for the power function of the next form : y=x^n,x∈N,n∈N to the numerical series. The most widely used ways 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

[17] **viXra:1512.0376 [pdf]**
*submitted on 2015-12-19 13:58:30*

**Authors:** Martin Erik Horn

**Comments:** 1 Page.

A formula for (1/7)!^42 = Gamma(8/7)^42 is given.

**Category:** Number Theory

[16] **viXra:1512.0368 [pdf]**
*submitted on 2015-12-18 12:14:14*

**Authors:** Hervé G.

**Comments:** 5 Pages.

An elementary and simple proof of the formula:
$\displaystyle 3\int_0^1\dfrac{1}{x}\arctan(\dfrac{x(1-x)}{2-x})dx=G$
$G$ being the Catalan's constant.

**Category:** Number Theory

[15] **viXra:1512.0363 [pdf]**
*submitted on 2015-12-18 09:51:28*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a strict sense, the term “prime quadruplet” refers strictly to the primes (p, p + 2, p + 6, p + 8) - see Wolfram MathWorld; it is not known if there are infinitely many such prime quadruplets. In this paper I conjecture that for any k non-null positive integer there exist an infinity of quadruplets of primes of the form (p, p+2k^2, p+6k^2, p+8k^2). Finally, I define the generalized Brun’s constant for prime quadruplets of the type showed and I estimate its value for the particular case k = 2 (for k = 1 the value it is known being approximately equal to 0.87).

**Category:** Number Theory

[14] **viXra:1512.0304 [pdf]**
*submitted on 2015-12-13 02:09:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any Carmichael number C is true one of the following two statements: (i) there exist at least one prime q, q lesser than Sqr (C), such that p = (C – q)/(q – 1) is prime, power of prime or semiprime m*n, n > m, with the property that n – m + 1 is prime or power of prime or n + m – 1 is prime or power of prime; (ii) there exist at least one prime q, q lesser than Sqr (C), such that p = (C – q)/((q – 1)*2^n) is prime or power of prime. In two previous papers I made similar assumptions on the squares of primes of the form 10k + 1 respectively 10k + 9 and I always believed that Fermat pseudoprimes behave in several times like squares of primes.

**Category:** Number Theory

[13] **viXra:1512.0301 [pdf]**
*submitted on 2015-12-12 13:51:39*

**Authors:** Hitesh Jain

**Comments:** 3 Pages.

To find Henry Dudeney’s Curious Numbers using pattern.

**Category:** Number Theory

[12] **viXra:1512.0291 [pdf]**
*replaced on 2015-12-24 22:01:04*

**Authors:** Quang Nguyen Van

**Comments:** 4 Pages.

There is now no a method for searching a prime number in the interval [ $ p_{m},p_{m+1}^{2} $] by using formula, and there is no known useful formula that sets apart all prime numbers from composites. In this paper, we try to use a formula for searching a prime number in the interval [ $ p_{m},p_{m+1}^{2} $], if a complete list of prime numbers up to $ p_{m} $ is known,and we also give an open problem on this formula.

**Category:** Number Theory

[11] **viXra:1512.0284 [pdf]**
*submitted on 2015-12-11 14:34:52*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any square of prime of the form p^2 = 10k + 1, p greater than or equal to 11, is true that there exist at least one prime q, q lesser than p, such that r = (p^2 – q)/(q – 1) is prime and, in case that this conjecture turns out not to be true, I considered three related “weaker” conjectures.

**Category:** Number Theory

[10] **viXra:1512.0265 [pdf]**
*submitted on 2015-12-08 06:59:58*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I conjecture that for any square of prime of the form p^2 = 10k + 9, p greater than or equal to 7, is true that there exist at least one prime q, q lesser than p, such that r = (p^2 – q)/(q – 1) is prime and, in case that this conjecture turns out not to be true, I considered three related “weaker” conjectures.

**Category:** Number Theory

[9] **viXra:1512.0255 [pdf]**
*submitted on 2015-12-06 18:17:45*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that the formula 9*m*n^3 + 3*n^3 – 15*m*n^2 + 6*m*n – 2*n^2 produces Poulet numbers, and I conjecture that this formula produces an infinite sequence of Poulet numbers for any m non-null positive integer.

**Category:** Number Theory

[8] **viXra:1512.0252 [pdf]**
*submitted on 2015-12-07 03:10:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that the formula 8*m*n^3 + 40*n^3 + 38*n^2 + 6*m*n^2 + m*n + 11*n + 1 produces Poulet numbers, and I conjecture that this formula produces an infinite sequence of Poulet numbers for any m non-null positive integer.

**Category:** Number Theory

[7] **viXra:1512.0250 [pdf]**
*submitted on 2015-12-06 15:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present three cubic polynomials that generate (probably infinite) sequences of Poulet numbers.

**Category:** Number Theory

[6] **viXra:1512.0244 [pdf]**
*submitted on 2015-12-06 11:56:03*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that the formula m*n^2 + 11*m*n – 23*n + 19*m - 49 produces Poulet numbers, and I conjecture that this formula produces an infinite sequence of Poulet numbers for any m non-null positive integer, respectively for any n non-null positive integer.

**Category:** Number Theory

[5] **viXra:1512.0242 [pdf]**
*submitted on 2015-12-06 06:56:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that the formula 4*n^2 + 37*n + 85 produces Poulet numbers, and I conjecture that this formula is generic for an infinite sequence of Poulet numbers.

**Category:** Number Theory

[4] **viXra:1512.0240 [pdf]**
*submitted on 2015-12-06 09:39:05*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that the formula 2*n^2 + 147*n + 2701 produces Poulet numbers, and I conjecture that this formula is generic for an infinite sequence of Poulet numbers.

**Category:** Number Theory

[3] **viXra:1512.0119 [pdf]**
*replaced on 2015-12-07 20:38:33*

**Authors:** Hajime Mashima

**Comments:** 15 Pages.

"6 or more natural number, it can be expressed as the sum of three prime numbers."
This is what you wrote in the letter to Christian Goldbach was addressed to Leonhard Euler in 1742.
Further, "6 or more even number is can be expressed by the sum of two odd primes." And it is equivalent.

**Category:** Number Theory

[2] **viXra:1512.0113 [pdf]**
*submitted on 2015-12-03 13:36:15*

Бесконечные ряды и Вселенная.

**Authors:** Putenikhin P.V.

**Comments:** 7 Pages. rus

In the literature on mathematics and cosmology can find arguments about the infinite, which is the erroneous conclusion that the part of infinity can be equal to the whole.

В литературе по математике и космологии можно встретить рассуждения о бесконечностях, в которых делается ошибочный вывод о том, что в бесконечности часть может быть равна целому.

**Category:** Number Theory

[1] **viXra:1512.0006 [pdf]**
*replaced on 2016-01-04 08:02:02*

**Authors:** Patrick Solé, Yuyang Zhu

**Comments:** 6 Pages.

The conjectured Robin inequality for an integer $n>7!$ is $\sigma(n)<e^\gamma n \log \log n,$ where $\gamma$ denotes Euler constant, and
$\sigma(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH).
Writing $D(n)=e^\gamma n \log \log n-\sigma(n),$ and $d(n)=\frac{D(n)}{n},$ we prove unconditionally that
$\liminf_{n \rightarrow \infty} d(n)=0.$
The main ingredients of the proof are an estimate for Chebyshev summatory function,
and an effective version of Mertens third theorem due to Rosser and Schoenfeld.
A new criterion for RH depending solely on $\liminf_{n \rightarrow \infty}D(n)$ is derived.

**Category:** Number Theory