[23] **viXra:1711.0417 [pdf]**
*replaced on 2017-11-25 21:37:21*

**Authors:** Liu Ran

**Comments:** 2 Pages.

A new way to prove prime number distribution being no law.

**Category:** Number Theory

[22] **viXra:1711.0353 [pdf]**
*submitted on 2017-11-19 03:41:41*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: Any square of a prime p^2, where p > 3, can be written as p + q + (n*q – n + 1) or as p + q + (n*q - n – 1), where q and n*q – n + 1 respectively n*q - n – 1 are primes and n positive integer. Examples: 11^2 = 121 = 11 + 37 + (2*37 – 1), where 37 and 2*37 – 1 = 73 are primes; 13^2 = 169 = 13 + 53 + (2*53 – 3), where 53 and 2*53 – 3 = 103 are primes. An equivalent formulation of the conjecture is that for any prime p, p > 3, there exist n positive integer such that one of the numbers q = (p^2 – p + n – 1)/(n + 1) or q = p^2 – p + n + 1)/(n + 1) is prime satisfying also the condition that p^2 – p – q is prime.

**Category:** Number Theory

[21] **viXra:1711.0343 [pdf]**
*submitted on 2017-11-18 03:29:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

Playing with Carmichael numbers, a set of numbers I’ve always been fond of (I’ve “discovered” Fermat’s “Little” Theorem and the first few Carmichael numbers before I know they had already been discovered), I noticed that the formula C + 81*2^(4*d), where C is a Carmichael number and d one of its prime factors, gives often primes or products of very few primes. For instance, for C = 1493812621027441 are obtained in this manner three primes: 2918779690625137, 6729216728661136606577017055290271857 and 644530914387083488233375393598279808770191171433362641802841314053534708129737067311868017 (a 90-digit prime!), respectively for d = 11, d = 29 and d = 73.

**Category:** Number Theory

[20] **viXra:1711.0330 [pdf]**
*submitted on 2017-11-17 01:34:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1 and dn be the least respectively the largest prime factors of the number P1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form P1 + |P1 – dn^2|*d1, where |P1 – dn^2| is the absolute value of P1 – dn^2. Example: for Poulet number P1 = 1729 = 7*13*19 is obtained through this operation Poulet number P2 = 11305 (1729 – 19^2 = 1368 and 1729 + 1368*7 = 11305). Note that from 11 from the first 30 Poulet numbers (P1) were obtained through this method Poulet numbers (P2).

**Category:** Number Theory

[19] **viXra:1711.0307 [pdf]**
*submitted on 2017-11-14 06:41:14*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some formulas involving pi and G (Catalan constant).

**Category:** Number Theory

[18] **viXra:1711.0303 [pdf]**
*submitted on 2017-11-14 06:51:50*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some elementary integrals for pi.

**Category:** Number Theory

[17] **viXra:1711.0296 [pdf]**
*replaced on 2018-02-01 05:05:33*

**Authors:** Kurmet Sultan

**Comments:** 15 Pages. 15

In this paper we give a brief proof of the Collatz conjecture. It is shown that it is more efficient to start calculating the Collatz function C (n) from odd numbers 6m ± 1. It is further proved that if we calculate by the formula ((6n ± 1)·2^q -1) / 3 on the basis of a sequence of numbers 6n ± 1, increasing the exponent of two by 1 at each iteration, then to each number of the form 6n ± 1 there will correspond a set whose elements are numbers of the form 3t, 6m-1 and 6m + 1. Moreover, all sets are disjoint. Then it is shown that if we construct micro graphs of numbers by combining the numbers 6n ± 1 with their elements of the set 3t, 6m-1 and 6m + 1, then combine the micro graphs by combining equal numbers 6n ± 1 and 6m ± 1, then a tree-like fractal graph of numbers. A tree-like fractal graph of numbers, each vertex of which corresponds to numbers of the form 6m ± 1, is a proof of the Collatz conjecture, since any of its vertices is connected with a finite vertex connected with unity.

**Category:** Number Theory

[16] **viXra:1711.0291 [pdf]**
*replaced on 2017-11-30 10:59:47*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages. This version adds an example using Leibniz tables.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.

**Category:** Number Theory

[15] **viXra:1711.0276 [pdf]**
*submitted on 2017-11-11 13:07:08*

**Authors:** Dariusz Dudało

**Comments:** 1 Page.

Monty Hall problem

**Category:** Number Theory

[14] **viXra:1711.0267 [pdf]**
*submitted on 2017-11-10 23:39:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: The square of any odd prime can be obtained from the numbers of the form 360*k + 72 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number 360*k + 72; than for any square of a prime p^2 there exist k such that (d1 - 1)*(d2 - 1)*...*(dn - 1) + 1 = p^2. Example: for p^2 = 13^2 = 169 there exist k = 17 such that from 360*17 + 72 = 6192 = 2^4*3^2*43 is obtained 1^4*2^2*42 + 1 = 169. I also conjecture that any absolute Fermat pseudoprime (Carmichael number) can be obtained through the presented formula, which attests again the special relation that I have often highlighted between the nature of Carmichael numbers and the nature of squares of primes.

**Category:** Number Theory

[13] **viXra:1711.0262 [pdf]**
*submitted on 2017-11-10 11:00:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number P1 – 1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form (d1 + 1)*(d2 + 1)*...*(dn + 1) + 1. Example: for Poulet number P1 = 645 is obtained through this operation Poulet number P2 = 1729 (644 = 2*2*7*23 and 3*3*8*24 + 1 = 1729). Note that from more than one Poulet number P1 can be obtained the same Poulet number P2 (from both 1729 and 6601 is obtained 46657).

**Category:** Number Theory

[12] **viXra:1711.0258 [pdf]**
*replaced on 2017-11-10 12:40:06*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages. A more complete bibliography is included.

This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.

**Category:** Number Theory

[11] **viXra:1711.0247 [pdf]**
*submitted on 2017-11-07 09:41:06*

**Authors:** Edigles Guedes

**Comments:** 14 Pages.

We demonstrate some elementary identities for quocient of q-series.

**Category:** Number Theory

[10] **viXra:1711.0239 [pdf]**
*submitted on 2017-11-07 03:53:46*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents two BBP-type formulas

**Category:** Number Theory

[9] **viXra:1711.0236 [pdf]**
*submitted on 2017-11-06 18:00:00*

**Authors:** Edigles Guedes

**Comments:** 16 Pages.

We demonstrate some elementary identities for q-series involving the q-Pochhammer symbol, as well as an identity involving the generating functions of the (m,k)-capsids and (m, r1, r2)-capsids.

**Category:** Number Theory

[8] **viXra:1711.0203 [pdf]**
*submitted on 2017-11-05 20:45:06*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds which belong not to AX+BY=CZ. Then, affirm AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, prove AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric law of odd numbers whereby even number 2W-1HZ as symmetric center after divide the inequality in four. Finally, reach 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

[7] **viXra:1711.0202 [pdf]**
*submitted on 2017-11-06 02:56:59*

**Authors:** Kunle Adegoke

**Comments:** 17 Pages.

We present new infinite arctangent sums and infinite sums of products of arctangents. Many previously known evaluations appear as special cases of the general results derived in this paper.

**Category:** Number Theory

[6] **viXra:1711.0140 [pdf]**
*submitted on 2017-11-04 16:02:17*

**Authors:** José de Jesús Camacho Medina

**Comments:** 3 Pages.

This article disseminates a series of new and interesting mathematical formulas for the fibonacci sequence as product of the investigations of the author since 2015.

**Category:** Number Theory

[5] **viXra:1711.0134 [pdf]**
*replaced on 2017-11-10 10:10:06*

**Authors:** Philip Gibbs, Judson McCranie

**Comments:** 9 Pages.

All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps.

**Category:** Number Theory

[4] **viXra:1711.0130 [pdf]**
*replaced on 2018-02-11 08:32:57*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages. Further simplifications and clarifications.

Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.

**Category:** Number Theory

[3] **viXra:1711.0128 [pdf]**
*submitted on 2017-11-03 22:24:48*

**Authors:** Choe Ryujin

**Comments:** 4 Pages.

Theorem of prime pairs

**Category:** Number Theory

[2] **viXra:1711.0127 [pdf]**
*submitted on 2017-11-03 23:29:58*

**Authors:** Bado idriss olivier

**Comments:** 7 Pages.

In this paperwe give the proof Polignac Conjecture
by using Chebotarev -Artin theorem ,Mertens formula and Poincaré sieve For doing that we prove that .Let's X be an arbitrarily large real number and n an even integer we prove that there are many primes p such that p+n is prime between sqrt(X) and X

**Category:** Number Theory

[1] **viXra:1711.0109 [pdf]**
*replaced on 2017-11-05 03:05:10*

**Authors:** Antoine Balan

**Comments:** 5 Pages.

We introduce a generalization of the q-calculus, which we call qq'-calculus. Some formulas are obtained; however the theory remains limited.

**Category:** Number Theory