Number Theory

1711 Submissions

[24] 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
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

[23] 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
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

[22] 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
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

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

Question 405 : pi and G

Authors: Edgar Valdebenito
Comments: 3 Pages.

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

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

Question 416 : Pi , Integral Representations

Authors: Edgar Valdebenito
Comments: 3 Pages.

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

[19] viXra:1711.0296 [pdf] replaced on 2017-11-17 23:40:31

A Brief Proof of the Collatz Conjecture

Authors: Kurmet Sultan
Comments: 9 Pages. Russian version

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

[18] viXra:1711.0291 [pdf] replaced on 2017-11-16 09:44:04

The Irrationality of Trigonometric and Hyperbolic Functions

Authors: Timothy W. Jones
Comments: 6 Pages. Clarifications of lemmas, slight re-organization.

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

[17] viXra:1711.0283 [pdf] submitted on 2017-11-12 23:30:43

Riemann Hypothesis Rendered as not Provable

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

A sentence to test is if known zeroes imply other zeroes. This effectively tests if a location of zeroes (trivial based on even numbers) and a location of zeroes (non trivial based on odd numbers) imply possibly another location of zeroes as a tautology, because the question is "Are there possibly other zeroes".
Category: Number Theory

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

Monty Hall Problem

Authors: Dariusz Dudało
Comments: 1 Page.

Monty Hall problem
Category: Number Theory

[15] 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
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

[14] 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
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

[13] 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

[12] viXra:1711.0249 [pdf] submitted on 2017-11-08 09:49:05

On Expansion of Convergence Domain of Dirichlet Series Determining the Riemann Zeta Function

Authors: I. N. Tukaev
Comments: 3 Pages.

This paper proves that the Dirichlet series determining the Riemann zeta function converges within a domain of a real component of complex variable equal to one, with an imaginary component non-equal to zero.
Category: Number Theory

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

Elementary Identities for Quocient of q -Series

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

BBP - High - Precision Arithmetic

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

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

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

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

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

Sums of Arctangents and Sums of Products of Arctangents

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

Interesting Formulas for the Fibonacci Sequence

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

The Ulam Numbers up to One Trillion

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 2017-11-09 06:45:07

The Irrationality and Transcendence of e Connected

Authors: Timothy W. Jones
Comments: 3 Pages. Slight corrections.

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

Theorem of Prime Pairs

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

Demonstration de la Conjecture de Polignac

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

The QQ'-Calculus

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