# Number Theory

## 1711 Submissions

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

### 素数分布无规律

Authors: Liu Ran

A new way to prove prime number distribution being no law.
Category: Number Theory

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

### Conjecture that Any Square of a Prime P^2 Can be Written as P+q+(nq-N±1) Where Q and nq-N±1 Primes

Authors: Marius Coman

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

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

### A Method of Obtaining Large Primes Based on Carmichael Numbers

Authors: Marius Coman

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

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

### A Set of Poulet Numbers Defined by an Interesting Relation Between Their Prime Factors

Authors: Marius Coman

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

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

### Question 405 : pi and G

Authors: Edgar Valdebenito

This note presents some formulas involving pi and G (Catalan constant).
Category: Number Theory

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

### Question 416 : Pi , Integral Representations

Authors: Edgar Valdebenito

This note presents some elementary integrals for pi.
Category: Number Theory

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

### Proof of the Collatz Conjecture

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

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

### The Irrationality of Trigonometric and Hyperbolic Functions

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

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

### Monty Hall Problem

Authors: Dariusz Dudało

Monty Hall problem
Category: Number Theory

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

### Conjecture that Any Square of a Prime Can be Obtained Through an Unusual Operation on the Numbers 360k+72

Authors: Marius Coman

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

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

### An Unusual Operation on a Set of Poulet Numbers Which Conducts to Another Set of Poulet Numbers

Authors: Marius Coman

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

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

### The Squared Case of Pi^n is Irrational Gives Pi is Transcendental

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

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

### Elementary Identities for Quocient of q -Series

Authors: Edigles Guedes

We demonstrate some elementary identities for quocient of q-series.
Category: Number Theory

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

### BBP - High - Precision Arithmetic

Authors: Edgar Valdebenito

This note presents two BBP-type formulas
Category: Number Theory

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

### Some Elementary Identities in Q-Series and the Generating Functions of the (M,k)-Capsids and (M, R1, R2)-Capsids

Authors: Edigles Guedes

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

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

### A Step -by- Step Proof of Beal’s Conjecture

Authors: Zhang Tianshu

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

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

### Sums of Arctangents and Sums of Products of Arctangents

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

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

### Interesting Formulas for the Fibonacci Sequence

Authors: José de Jesús Camacho Medina

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

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

### The Ulam Numbers up to One Trillion

Authors: Philip Gibbs, Judson McCranie

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

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

### The Irrationality and Transcendence of e Connected

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

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

### Theorem of Prime Pairs

Authors: Choe Ryujin

Theorem of prime pairs
Category: Number Theory

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

### Demonstration de la Conjecture de Polignac

Authors: Bado idriss olivier