Number Theory

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2007 - 0703(3) - 0706(2)
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2012 - 1201(2) - 1202(13) - 1203(7) - 1204(9) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(15) - 1211(10) - 1212(4)
2013 - 1301(5) - 1302(10) - 1303(16) - 1304(15) - 1305(12) - 1306(13) - 1307(26) - 1308(12) - 1309(9) - 1310(13) - 1311(16) - 1312(21)
2014 - 1401(20) - 1402(11) - 1403(25) - 1404(13) - 1405(17) - 1406(21) - 1407(35) - 1408(52) - 1409(47) - 1410(21)

Recent submissions

Any replacements are listed further down

[822] viXra:1410.0174 [pdf] submitted on 2014-10-27 11:29:15

Solving Diophantine Equations

Authors: Octavian Cira, Florentin Smarandache
Comments: 252 Pages.

In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation η(π(x)) = π(η(x)), where η is the Smarandache function and π is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful ”tools” for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.
Category: Number Theory

[821] viXra:1410.0142 [pdf] submitted on 2014-10-23 05:19:00

Invariant Relations Between Binary Goldbach’s Decompositions’numbers Coded in a 4 Letters Language

Authors: Denise Vella-Chemla
Comments: 23 Pages.

We propose a modelization of binary Goldbach's decompositions in a 4 letters language that permits to envisage this problem in a new way.
Category: Number Theory

[820] viXra:1410.0140 [pdf] submitted on 2014-10-23 05:48:34

Two Exciting Classes of Odd Composites Defined by a Relation Between Their Prime Factors

Authors: Marius Coman
Comments: 3 Pages.

In this paper I will define two interesting classes of odd composites often met (by the author of this paper) in the study of Fermat pseudoprimes, which might also have applications in the study of big semiprimes or in other fields. This two classes of composites n = p(1)*...*p(k), where p(1), ..., p(k) are the prime factors of n are defined in the following way: p(j) – p(i) + 1 is a prime or a power of a prime, respectively p(i) + p(j) – 1 is a prime or a power of prime for any p(i), p(j) prime factors of n such that p(1) ≤ p(i) < p(j) ≤ p(k).
Category: Number Theory

[819] viXra:1410.0120 [pdf] submitted on 2014-10-21 08:20:36

Goldbach´s Conjecture Implies Twin Prime Conjecture

Authors: O.Emilio.C.Sánchez
Comments: 2 Pages.

These two conjectures are perhaps the two most famous unsolved problems in number theory but, in fact, they are closely linked. In this concise article we will prove that Goldbach´s conjecture implies twin prime´s. Whether twin prime´s conjecture implies Goldbach´s conjecture (and then both conjectures would became equivalent) or not, will be matter of further and hard working.
Category: Number Theory

[818] viXra:1410.0114 [pdf] submitted on 2014-10-19 17:08:37

The Nonexistence of Odd Perfect Numbers

Authors: A. Garcés Doz
Comments: 7 Pages.

This proof uses a congruence, which is implicit in the condition, mandatory, demonstrated by Euler. More precisely, a congruence that must be fulfilled in the equation that equals the odd number N, with Euler condition, and the formula for the sum of the divisors of the number N. Following a rigorous and meticulous way, this mandatory congruence; a final equation is obtained after several polynomials simplifications on both sides of the original equation that equals the number 2N with the sum of the divisors of the number N. With this final equation, the impossibility of the existence of odd perfect numbers is demonstrated by the impossibility of the equation \;2N=\sigma(N)\; is fulfilled.
Category: Number Theory

[817] viXra:1410.0112 [pdf] submitted on 2014-10-19 23:08:56

An Approach to Explore the Infinite Nature of Twin Primes

Authors: Prashanth R. Rao
Comments: 2 Pages.

Abstract: Twin prime conjecture states that there are infinite number of twin primes of the form p and p+2. Remarkable progress has recently been achieved by Y. Zhang to show that infinite primes that differ by large gap (~ 70 million) exist and this gap has been further narrowed to ~600 by others. We use an elementary approach to explore any obvious constraint that could limit the infinite nature of twin primes. Using Fermat’s little theorem as a surrogate for primality we derive an equation that suggests but not prove that twin primes can be infinite.
Category: Number Theory

[816] viXra:1410.0108 [pdf] submitted on 2014-10-19 11:45:01

A Simple Criterion for Finding Mersenne Primes

Authors: Zeraoulia Elhadj
Comments: 2 Pages.

In this note, we introduce a simple criterion to prove that a given Mersenne number is really a Mersenne prime.
Category: Number Theory

[815] viXra:1410.0107 [pdf] submitted on 2014-10-19 06:16:32

A Proof of the ABC Conjecture

Authors: Zhang Tianshu
Comments: 16 Pages.

We first get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. Next expound relations between C and paf (ABC) by the symmetric law of odd numbers. Finally we have proven C ≤ Cε [paf (ABC)] 1+ ε such being the case A+B=C, and gcf (A, B, C) =1.
Category: Number Theory

[814] viXra:1410.0068 [pdf] submitted on 2014-10-13 07:58:09

Prime Number Symmetrie Theorem

Authors: Maik Becker-Sievert
Comments: 1 Page.

Every Integer stands in the center of two Integer-Primes
Category: Number Theory

[813] viXra:1410.0066 [pdf] submitted on 2014-10-13 04:16:43

Notes On the Proof of Second Hardy-Littlewood Conjecture

Authors: S. Roy
Comments: 4 Pages.

In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture, namely the inequality $\pi(x) + \pi(y) > \pi(x + y)$ is examined, where $\pi(x)$ denotes the number of primes not exceeding $x$. It is shown that there the inequalty holds for all suciently large $x$ and $y$.
Category: Number Theory

[812] viXra:1410.0059 [pdf] submitted on 2014-10-12 09:18:30

A Study of Ralationship of RSA with Goldbach’s Conjecture and It’s Properties

Authors: Chenglian LIU, Jian YE
Comments: 17 Pages.

The Goldbach's conjecture has plagued mathematicians for over two hundred and seventy years. Whether a professional or an amateur enthusiast, all have been fascinated by this question. Why do mathematicians have no way to solve this problem? Up until now, Chen has been recognized for the most concise proof his “1 + 2” theorem in 1973. In this article, the author will use elementary concepts to describe and indirectly prove the Goldbach conjecture.
Category: Number Theory

[811] viXra:1410.0055 [pdf] submitted on 2014-10-11 20:50:39

Riemann Hypothesis and Primorial Number

Authors: Choe Ryong Gil
Comments: 8 pages, 2 tables

In this paper we consider the Riemann hypothesis by the primorial numbers.
Category: Number Theory

[810] viXra:1410.0054 [pdf] submitted on 2014-10-12 03:43:38

The Art of Inspired Guessing

Authors: Simon Plouffe
Comments: 5 Pages.

A presentation of various formulas is given. Many of these findings have no explanation whatsoever. One is related to the mass ratio of the neutron and proton: 1.00137841917. They were found using a variety of methods using either a HP‐15C calculator in 1988 to the current database of constants of the author which consist of 12.3 billion entries.
Category: Number Theory

[809] viXra:1410.0042 [pdf] submitted on 2014-10-10 02:03:45

Conjecture that States that a Mersenne Number with Odd Exponent is Either Prime Either Divisible by a 2-Poulet Number

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make a conjecture which states that any Mersenne number (number of the form 2^n – 1, where n is natural) with odd exponent n, where n is greater than or equal to 3, also n is not a power of 3, is either prime either divisible by a 2-Poulet number. I also generalize this conjecture stating that any number of the form P = ((2^m)^n – 1)/3^k, where m is non-null positive integer, n is odd, greater than or equal to 5, also n is not a power of 3, and k is equal to 0 or is equal to the greatest positive integer such that P is integer, is either a prime either divisible by at least a 2-Poulet number (I will name this latter numbers Mersenne-Coman numbers) and I finally enunciate yet another related conjecture.
Category: Number Theory

[808] viXra:1410.0041 [pdf] submitted on 2014-10-10 02:04:05

Conjecture that States that a Fermat Number is Either Prime Either Divisible by a 2-Poulet Number

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make a conjecture which states that any Fermat number (number of the form 2^(2^n) + 1, where n is natural) is either prime either divisible by a 2-Poulet number. I also generalize this conjecture stating that any number of the form N = ((2^m)^p + 1)/3^k, where m is non-null positive integer, p is prime, greater than or equal to 7, and k is equal to 0 or is equal to the greatest positive integer such that N is integer, is either a prime either divisible by at least a 2-Poulet number (I will name this latter numbers Fermat-Coman numbers) and I finally enunciate yet another related conjecture.
Category: Number Theory

[807] viXra:1410.0038 [pdf] submitted on 2014-10-09 04:48:20

A Modification of Riesel Primality Test

Authors: Predrag Terzic
Comments: 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k2^n-1 is introduced .
Category: Number Theory

[806] viXra:1410.0030 [pdf] submitted on 2014-10-07 09:43:02

The Generalized Goldbach's Conjecture:the Corollary of Symmetry Primes

Authors: Jian Ye
Comments: 3 Pages.

Goldbach’s conjecture: symmetry primes exists in natural numbers.the generalized Goldbach’s conjecture: symmetry of primes in arithmetic progression still exists.
Category: Number Theory

[805] viXra:1410.0025 [pdf] submitted on 2014-10-05 16:25:44

Unsolved Problem in Analysis :On Irrationlity of 2^e

Authors: Md Zahid
Comments: 5 Pages.

2^e is rational or irrational number is not known. It is unsolved and open problem in analysis [1] .In this paper we proved that 2^e as irrational number. We attack the proof by method of contradiction. We assume that 2^e be rational number. Then we use some logarithms properties and simplification to get a relation between ‘e’ and assumed rational number, since we known that ‘e’ is irrational number, we use some further simplification and method to prove that 2^e is irrational number .
Category: Number Theory

[804] viXra:1410.0017 [pdf] submitted on 2014-10-04 07:44:59

A Proof of the Beal's Conjecture (After Third Modification)

Authors: Zhang Tianshu
Comments: 22 Pages.

First we classify A, B and C according to their respective odevity, and ret rid of two kinds from AX+BY=CZ. Then affirm AX+BY=CZ such being the case A, B and C have a common prime factor by examples. After that, prove AX+BY≠CZ under these circumstances that A, B and C have not any common prime factor by mathematical analyses with the aid of the symmetric law of odd numbers. Finally we have proven that the Beal’s conjecture holds water after the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.
Category: Number Theory

[803] viXra:1410.0011 [pdf] submitted on 2014-10-03 02:39:32

Prime Number Theorem

Authors: Maik Becker-Sievert
Comments: 1 Page.

For every odd prime number exist a sum (x+y) so that (x-y) is also a prime number. Every odd number is the difference of two square numbers Every 4n number is the difference of two square numbers
Category: Number Theory

[802] viXra:1409.0167 [pdf] submitted on 2014-09-24 04:26:27

Fermat and Mersenne Prime Criteria for the Infinity or the Strong Finiteness of Primes of the Form 2^x±1

Authors: Pingyuan Zhou
Comments: 29 Pages. Author pesents a mathematical partten in which Fermat and Mersenne primes can become criteria for the infinity or the strong finiteness each other.

Abstract: This paper presents that Fermat primes and Mersenne primes can separately become criteria for the infinity or the strong finiteness of primes of the form 2^x±1, which includes Fermat prime criteria for the set of Mersenne primes and its two subsets as well as Mersenne prime criteria for the set of Fermat primes and its two subsets.
Category: Number Theory

[801] viXra:1409.0166 [pdf] submitted on 2014-09-24 05:00:01

Fermat Prime Criterion Related to Landau's Fourth Problem

Authors: Pingyuan Zhou
Comments: 9 Pages. Anthor gives an argument for solving Landau's fourth problem from a hypothesis on the infinity of primes represented by quadratic polynomial of Mersenne primes.

Abstract: In this paper we consider the infinity of primes represented by quadratic polynomial with 4(Mp-2)^2+1, basing on a hypothesis as sufficient condition in which Fermat primes are criterion for the infinity of such primes, where Mp is Mersenne prime, and give an elementary argument for existence of infinitely many primes of the form x^2+1. As an addition, an elementary argument on the infinity of Mersenne primes is also given.
Category: Number Theory

[800] viXra:1409.0156 [pdf] submitted on 2014-09-22 03:45:29

Primality Tests for Specific Classes of Proth Numbers

Authors: Predrag Terzic
Comments: 3 Pages.

Polynomial time primality tests for specific classes of Proth numbers are introduced .
Category: Number Theory

[799] viXra:1409.0155 [pdf] submitted on 2014-09-21 16:19:34

The Sum of the Digits of a Number/primality Testing

Authors: Ounas Meriam
Comments: 04 Pages.

In this paper, I will try to explain my idea about the world of the digits of numbers which is somewhat circumvented by mathematicians.
Category: Number Theory

[798] viXra:1409.0151 [pdf] submitted on 2014-09-21 03:37:44

Presentation of Plouffe's Inverter, L'Inverseur de Plouffe

Authors: Simon Plouffe
Comments: 23 Pages. the paper is in french

A presentation (1998) is made of Plouffe's Inverter, a conference in march 1998 at the Université du Québec À Montréal.
Category: Number Theory

[797] viXra:1409.0144 [pdf] submitted on 2014-09-18 03:01:00

Perfect Cuboid Does not Exist

Authors: Saenko V.I.
Comments: Pages. The proof should be improved because in the present form it is valid only if all non-unit G_i are prime.

A perfect cuboid, i.e., a rectangular parallelepiped having integer edges, integer face diagonals, and integer space diagonal, is proved to be non-existing.
Category: Number Theory

[796] viXra:1409.0135 [pdf] submitted on 2014-09-16 21:58:47

The Construction of Certain Numbers with Ruler and Compass, le Calcul de Certains Nombres Avec la Règle et le Compas

Authors: Simon Plouffe
Comments: 8 Pages. The construction of certain numbers with ruler and compass

Conference in french in Montréal in 1998 about the construction of arctan(1/2)/Pi and other numbers.
Category: Number Theory

[795] viXra:1409.0111 [pdf] submitted on 2014-09-13 15:43:12

The 400 Billion’th Binary Digit of pi is 0

Authors: Simon Plouffe
Comments: 16 Pages. talk in Ottawa 1997.

A talk given in Ottawa in 1997 about the computation of pi in binary.
Category: Number Theory

[794] viXra:1409.0110 [pdf] submitted on 2014-09-13 16:06:52

Miscellaneous Mathematical Constants

Authors: Simon Plouffe
Comments: 96 Pages.

This is my collection of mathematical constants evaluated to many digits. The document was given a copy to the gutenberg project in 1996.
Category: Number Theory

[793] viXra:1409.0108 [pdf] submitted on 2014-09-13 16:13:26

The first 498 Bernoulli Numbers

Authors: Simon Plouffe
Comments: 23 Pages.

A list of the first 498 Bernoulli Numbers. This text was published in 1996 and donated to the Gutenberg Project.
Category: Number Theory

[792] viXra:1409.0107 [pdf] submitted on 2014-09-13 16:14:18

The first 1000 Euler Numbers

Authors: Simon Plouffe
Comments: 112 Pages.

A list of the first 1000 Euler Numbers. This text was published in 1996 and donated to the Gutenberg Project.
Category: Number Theory

[791] viXra:1409.0106 [pdf] submitted on 2014-09-13 16:14:58

The first 1000 Fibonacci Numbers

Authors: Simon Plouffe
Comments: 38 Pages.

A list of the first 1000 Fibonacci Numbers. This text was published in 1996 and donated to the Gutenberg Project.
Category: Number Theory

[790] viXra:1409.0103 [pdf] submitted on 2014-09-13 14:48:21

How to Guess a Generating Function

Authors: Simon Plouffe
Comments: 24 Pages. bitmaps from hypercard stack

Conference given in Vancouver in 1995 at Simon Fraser University. keywords : generating function, GFUN, Encyclopedia of integer sequences, sequence, rational polynomial
Category: Number Theory

[789] viXra:1409.0101 [pdf] submitted on 2014-09-13 03:44:58

Compositeness Tests for Specific Classes of K3^n+2

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness tests for specific classes of numbers of the form k3^n+2 are introduced .
Category: Number Theory

[788] viXra:1409.0100 [pdf] submitted on 2014-09-12 21:15:51

The Lattice Reduction Algorithm and Applications, LLL and PSLQ

Authors: Simon Plouffe
Comments: 29 Pages.

Conference in 1996 at SFU Vancouver and Montréal. I present a serie of examples using the LLL algorithm.
Category: Number Theory

[787] viXra:1409.0099 [pdf] submitted on 2014-09-12 21:22:49

A Search for a Mathematical Expression for Mass Ratios Using a Large Database

Authors: Simon Plouffe
Comments: 15 Pages.

A computation experiment was conducted on mass ratio of fundamental particles. A series of method are explained.The interest is on the methodology used. The goal was to verify that no simple answer exist yet.
Category: Number Theory

[786] viXra:1409.0098 [pdf] submitted on 2014-09-12 21:25:41

Les Empilements D'hexagone et Quelques Méthodes

Authors: Simon Plouffe
Comments: 26 Pages. This is a talk made in Montréal in 1992.

En physique on modélise le comportement des gaz, du cristal de glace et du ferromagnétisme par l'étude de l'empilement d'objets sur le plan Z*Z, ou dans l'espace. On tente d'expliquer surtout les phénomènes de transition de phase. Ce qui intéresse les physiciens c'est le comportement du système lorsque la température T tend vers Tc, une température à laquelle se fait la transition entre deux états. Si on a une formule explicite on peut simuler pour de grandes valeurs. On compte, en prenant modèle sur les partitions ordinaires. L'énergie d'interaction des molécules entre elles étant comptée comme une "arête" entre 2 sommets i et j du plan Z*Z. L'interaction se mesure alors avec 2 variables qu'on somme sur toutes les positions possibles. On cherche donc la limite quand N tend vers l'infini. (N grand : beaucoup de molécules).
Category: Number Theory

[785] viXra:1409.0095 [pdf] submitted on 2014-09-12 10:28:07

Computing the Generating Function of a Series Given Its First Few Terms

Authors: Simon Plouffe, François Bergeron
Comments: 6 Pages. Published in 1991

We outline an approach for the computation of a good candidate for the generating function of a power series for which only the first few coefficients are known. More precisely, if the derivative, the logarithmic derivative, the reversion, or another transformation of a given power series (even with polynomial coefficients) appears to admit a rational generating function, we compute the generating function of the original series by applying the inverse of those transformations to the rational generating function found.
Category: Number Theory

[784] viXra:1409.0094 [pdf] submitted on 2014-09-12 10:29:19

A Relative of the Thue-Morse Sequence

Authors: Simon Plouffe, Jean-Paul Allouche, André Arnold, Srecko Brlek, Jean Berstel, William Jockusch, Bruce E. Sagan
Comments: 10 Pages. Published in 1992

We study a sequence, c, which encodes the lengths of blocks in the Thue-Morse sequence. In particular, we show that the generating function for c is a simple product.
Category: Number Theory

[783] viXra:1409.0093 [pdf] submitted on 2014-09-12 10:31:59

On the Rapid Computation of Various Polylogarithmic Constants

Authors: Simon Plouffe, David H. Bailey, Peter Borwein
Comments: 14 Pages. Published in 1997

We give an algorithm for the computation of the d'th digit of certain numbers in various bases.
Category: Number Theory

[782] viXra:1409.0092 [pdf] submitted on 2014-09-12 10:33:03

Recognizing Numerical Constants

Authors: Simon Plouffe, David H. Bailey
Comments: 17 Pages. Published in 1996

The advent of inexpensive, high-performance computers and new efficient algorithms have made possible the automatic recognition of numerically computed constants. In other words, techniques now exist for determining, within certain limits, whether a computed real or complex number can be written as a simple expression involving the classical constants of mathematics. These techniques will be illustrated by discussing the recognition of Euler sum constants, and also the discovery of new formulas for π and other constants, formulas that permit individual digits to be extracted from their expansions.
Category: Number Theory

[781] viXra:1409.0091 [pdf] submitted on 2014-09-12 10:34:44

The Quest for Pi

Authors: Simon Plouffe, David Bailey, Jon Borwein, Peter Borwein
Comments: 16 Pages. Published in 1997

This article gives a brief history of the analysis and computation of the mathematical constant π = 3.14159 . . ., including a number of the formulas that have been used to compute π through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of π to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of π to be computed.
Category: Number Theory

[780] viXra:1409.0089 [pdf] submitted on 2014-09-12 03:42:31

Compositeness Tests for Specific Classes of K3^n-2

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness tests for specific classes of numbers of the form k3^n-2 are introduced .
Category: Number Theory

[779] viXra:1409.0083 [pdf] submitted on 2014-09-11 13:03:50

An Efficient Algorithm for the Computation of Bernoulli Numbers

Authors: Simon Plouffe, Greg Fee
Comments: 8 Pages.

This article gives a direct formula for the computation of B (n) using the asymptotic formula
Category: Number Theory

[778] viXra:1409.0082 [pdf] submitted on 2014-09-11 13:05:32

Une Méthode Pour Obtenir la Fonction Génératrice D'une Série

Authors: Simon Plouffe
Comments: 11 Pages. Conference in Florence in 1993

Nous décrivons ici une méthode expérimentale permettant de calculer de bons candidats pour une forme close de fonctions génératrices à partir des premiers termes d’une suite de nombres rationnels.
Category: Number Theory

[777] viXra:1409.0081 [pdf] submitted on 2014-09-11 13:07:48

Approximations de Séries Génératrices et Quelques Conjectures

Authors: Simon Plouffe
Comments: 550 Pages. Master thesis 1992

master thesis of 1992, université du québec à Montréal. The thesis served as a template for the Encyclopedia of Integer Sequences in 1995 by Neil Sloane and Simon Plouffe
Category: Number Theory

[776] viXra:1409.0080 [pdf] submitted on 2014-09-11 13:10:25

On the Computation of the Nth Decimal Digit of Various Transcendental Numbers

Authors: Simon Plouffe
Comments: 8 Pages. Article of November 1996

A method for computing the n'th decimal digit of Pi in O(n^3log(n)^3) in time and with very little memory is presented here.
Category: Number Theory

[775] viXra:1409.0079 [pdf] submitted on 2014-09-11 13:12:12

Identities Inspired by the Ramanujan Notebooks, First Series

Authors: Simon Plouffe
Comments: 4 Pages.

I present here a collection of formulas inspired from the Ramanujan Notebooks. These formulas were found using an experimental method based on three widely available symbolic computation programs: PARI-Gp, Maple and Mathematica. A new formula is presented for Zeta(5) Une collection de formules inspirées des Notebooks de S. Ramanujan, elles ont toutes été trouvées par des méthodes expérimentales. Ces programmes de calcul symbolique sont largement disponibles (Pari-GP, Maple, Mathematica). Une nouvelle formule pour Zeta(5) est présentée.
Category: Number Theory

[774] viXra:1409.0078 [pdf] submitted on 2014-09-11 13:14:44

Identities Inspired by the Ramanujan Notebooks, Second Series

Authors: Simon Plouffe
Comments: 9 Pages.

A series of formula is presented that are all inspired by the Ramanujan Notebooks [6]. One of them appears in the notebooks II which is for Zeta(3). That formula inspired others that appeared in 1998, 2006 and 2009 on the author’s website and later in literature [1][2][3]. New formulas for and the Catalan constant are presented along with a surprising series of approximations. A new set of identities is given for Eisenstein series. All of the formulas are conjectural since they were found experimentally. A new method is presented for the computation of the partition function. Une série de formules utilisant l’exponentielle est présentée. Ces résultats reprennent ceux apparaissant en 1998, 2006 et 2009 sur [1][2][3]. Elles sont toutes inspirées des Notebooks de Ramanujan tels que Zeta(3). Une nouvelle série pour Zeta(3) et la constante de Catalan sont présentés ainsi qu’une série d’approximations surprenantes. Une série d’identités nouvelles sont présentées concernant les séries d’Eisenstein. Toutes les formules présentées sont des conjectures, elles ont toutes été trouvées expérimentalement. Une nouvelle méthode est présentée pour le calcul des partages d’un entier.
Category: Number Theory

[773] viXra:1409.0076 [pdf] submitted on 2014-09-11 11:10:16

Compositeness Test for Repunits Base 3

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness test for numbers of the form (3^p-1)/2 is introduced .
Category: Number Theory

[772] viXra:1409.0074 [pdf] submitted on 2014-09-11 03:52:26

A Proof of the Collatz Conjecture (After Second Modification)

Authors: Zhang Tianshu
Comments: 18 Pages. This is third manuscript for the article.

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 by operations on the contrary of the set operational rule for infinite many times. In this article, we will apply the mathematical induction with the help of certain operations by each other’s- opposed operational rules to prove that the Collatz conjecture is tenable.
Category: Number Theory

[771] viXra:1409.0073 [pdf] submitted on 2014-09-11 04:04:19

A Proof of the Beal's Conjecture (After Second Modification)

Authors: Zhang Tianshu
Comments: 24 Pages. This is third manuscript for the article.

In this article, we first have proven a lemma of EP+FV≠2M. Successively have proven the Beal’s conjecture by mathematical analyses with the aid of the lemma, such that enable the Beal’s conjecture holds water.
Category: Number Theory

[770] viXra:1409.0067 [pdf] submitted on 2014-09-10 00:52:06

Conjectured Compositeness Tests for Specific Classes of B^n-B+1 and B^n+b-1

Authors: Predrag Terzic
Comments: 2 Pages.

Compositeness criteria for specific classes of numbers of the form b^n-b+1 and b^n+b-1 are introduced .
Category: Number Theory

[769] viXra:1409.0064 [pdf] submitted on 2014-09-10 03:45:11

The Computation of Certain Numbers Using a Ruler and Compass

Authors: Simon Plouffe
Comments: 4 Pages.

We present a method for computing some numbers bit by bit using only a ruler and compass, and illustrate it by applying it to arctan(X)/π. The method is a spigot algorithm and can be applied to numbers that are constructible over the unit circle and the ellipse. The method is precise enough to produce about 20 bits of a number, that is, 6 decimal digits in a matter of minutes. This is surprising, since we do no actual calculations.
Category: Number Theory

[768] viXra:1409.0055 [pdf] submitted on 2014-09-09 03:28:17

Conference on Bernoulli Numbers, Montréal , Uqam 2003 (French)

Authors: Simon Plouffe
Comments: 11 Pages. an ascii version of a drawing of Pi by Yves Chiricota is given

A conference on Bernoulli numbers, a result is given on the Agoh‐Giuga conjecture, it has been verified up to n=49999. Also a formula on the sum of the fractional part of Bernoulli numbers and a sample session to the Inverter (Plouffe's Inverter) from a Maple session and results.
Category: Number Theory

[767] viXra:1409.0052 [pdf] submitted on 2014-09-07 20:57:00

When π(n) Divides N and When it Does not

Authors: Germán Paz
Comments: 6 Pages. Main text in English; abstract in English and Spanish. /// Texto principal en inglés; resumen en inglés y en español.

Let $\pi(n)$ denote the prime-counting function. In this paper we work with explicit formulas for $\pi(n)$ that are valid for infinitely many positive integers $n$, and we prove that if $n\ge 60184$ and $\operatorname{frac}(\ln n)=\ln n-\lfloor\ln n\rfloor>0.5$, then $\pi(n)$ does not divide $n$. Based on this result, we show that if $e$ is the base of the natural logarithm, $a$ is a fixed integer $\ge 11$ and $n$ is any integer in the interval $[e^{a+0.5},e^{a+1}]$, then $\pi(n)\nmid n$. In addition, we prove that if $n\ge 60184$ and $n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\ln n-1\rfloor$ located in the interval $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.

///////////////////

Sea $\pi(n)$ la función contadora de números primos. En este documento trabajamos con funciones explícitas para $\pi(n)$ que son válidas para infinitos enteros positivos $n$, y demostramos que si $n\ge 60184$ y $\operatorname{frac}(\ln n)=\ln n-\lfloor\ln n\rfloor>0.5$, entonces $\pi(n)$ no divide a $n$. Basándonos en este resultado, probamos que si $e$ es la base del logaritmo natural, $a$ es un entero fijo $\ge 11$ y $n$ es cualquier entero en el intervalo $[e^{a+0.5},e^{a+1}]$, entonces $\pi(n)\nmid n$. Además, demostramos que si $n\ge 60184$ y $n/\pi(n)$ es entero, entonces $n$ es un múltiplo de $\lfloor\ln n-1\rfloor$ ubicado en el intervalo $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.
Category: Number Theory

[766] viXra:1409.0048 [pdf] submitted on 2014-09-07 11:22:05

Conjectures from the Oeis Database, a Collection of 148403 Formulas

Authors: Simon Plouffe
Comments: 2562 Pages.

Conjectured formulas of the OEIS by Simon Plouffe as of Sept 6. 2014 There are 45691 unique sequence and more than 148403 expressions. Score = log(# of terms)*(length of sequence in charaters)/(length of the formula in characters).
Category: Number Theory

[765] viXra:1409.0045 [pdf] submitted on 2014-09-07 03:30:49

The Reflection of Light Rays in a Cup of Coffee

Authors: Simon Plouffe
Comments: 12 Pages. based of works done in 1974-1979 by Simon Plouffe

Analysis is made of the reflection of sunlight in a cup of coffee and how to obtain the same with congruences and prime numbers. Congruences, light rays, primitive roots, trigonometric sums, hypocycloids, epicycloids, binary expansion, nary expansion of 1/p.
Category: Number Theory

[764] viXra:1409.0044 [pdf] submitted on 2014-09-07 03:40:55

Exact Formulas for Integer Sequences

Authors: Simon Plouffe
Comments: 2 Pages.

A series of formulas are presented that permits the computation of the n'th term using the author customized bootstrap method. That method is a variant of what is described in [GKP]. The { } denotes the nearest integer function and [ ] the floor function. They were found in 1993. Annnnnn refers to either [Sloane] or [Sloane,Plouffe].
Category: Number Theory

[763] viXra:1409.0039 [pdf] submitted on 2014-09-06 07:53:48

One Hundred and Fifty Conjectures on Primes, Many Based on the Observation of Fermat Pseudoprimes

Authors: Marius Coman
Comments: 87 Pages.

In two of my previous published books, “Two hundred conjectures and one hundred and fifty open problems on primes”, respectively “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, I already expressed my conviction that the study of Fermat pseudoprimes, fascinating numbers that seem to be a little bit more willing to let themselves ordered and understood than the prime numbers, can help a lot in understanding these latter. This book brings together thirty-eight papers on prime numbers, many of them supporting the author’s belief, expressed above, namely that new ordered patterns can be discovered in the “undisciplined” set of prime numbers, observing the ordered patterns in the set of Fermat pseudoprimes, especially in the set of Carmichael numbers, the absolute Fermat pseudoprimes, and in the set of Poulet (sometimes also called Sarrus) numbers, the relative Fermat pseudoprimes to base two. Few papers, which are not based on the observation of pseudoprimes, though apparently heterogenous, still have something in common: they are all directed toward the same goal, discovery of new patterns in the set of primes, using the same means, namely the old and reliable integers. Part One of this book of collected papers contains one hundred and fifty conjectures on primes and Part Two of this book brings together the articles regarding primes, submitted by the author to the preprint scientific database Vixra, representing the context of the conjectures listed in Part One.
Category: Number Theory

[762] viXra:1409.0037 [pdf] submitted on 2014-09-06 03:51:34

An Interesting Relation Between the Squares of Primes and the Number 96 and Two Conjectures

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make two conjectures based on the observation of an interesting relation between the squares of primes and the number 96.
Category: Number Theory

[761] viXra:1409.0035 [pdf] submitted on 2014-09-06 04:18:22

A Formula that Seems to Generate Easily Big Numbers that Are Primes or Products of Very Few Primes

Authors: Marius Coman
Comments: 2 Pages.

The formula N = (p^4 – 2*p^2 + m)/(m – 1), where p is an odd prime and m is a positive integer greater than 1, seems to generate easily primes or products of very few primes.
Category: Number Theory

[760] viXra:1409.0034 [pdf] submitted on 2014-09-05 18:35:06

Four Conjectures Based on the Observation of a Type of Recurrent Sequences Involving Semiprimes

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make four conjectures starting from the observation of the following recurrent relations: (((p*q – p)*2 – p)*2 – p)...), respectively (((p*q – q)*2 – q)*2 – q)...), where p, q are distinct odd primes.
Category: Number Theory

[759] viXra:1409.0032 [pdf] submitted on 2014-09-05 16:10:03

Statements on the Infinity of Few Sequences or Types of Duplets or Triplets of Primes

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make few statements on the infinity of few sequences or types of duplets and triplets of primes which, though could appear heterogenous, are all based on the observation of the prime factors of absolute Fermat pseudoprimes, Carmichael numbers, or of relative Fermat pseudoprimes to base two, Poulet numbers.
Category: Number Theory

[758] viXra:1409.0028 [pdf] submitted on 2014-09-04 17:56:13

The Proof for Non-existence of Perfect Cuboid

Authors: Bambore Dawit
Comments: 9 Pages. the proof is short cut, there are instructions and results

This paper shows the non-existence of perfect cuboid by using two tools, the first is representing Pythagoras triplets by two numbers and the second is realizing the impossibility of two similar equations for the same problem at the same time in different ways and the variables of one is relatively less than the other. When we express all Pythagoras triplets in perfect cuboid problem and rearrange it we can get a single equation that can express perfect cuboid. Unfortunately perfect cuboid has more than two similar equations that can express it and contradict one another.
Category: Number Theory

[757] viXra:1409.0005 [pdf] submitted on 2014-09-02 02:56:46

Compositeness Test for Repunit Numbers

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time compositeness test for numbers of the form (10^n-1)/9 is introduced .
Category: Number Theory

[756] viXra:1409.0003 [pdf] submitted on 2014-09-01 10:02:24

无穷大的运算法则

Authors: Liu Ran
Comments: 1 Page.

传统数论中的无穷大是没有上界的,也就是没有最大,只有更大。无穷大是自相矛盾的。
Category: Number Theory

[755] viXra:1408.0231 [pdf] submitted on 2014-08-31 12:01:39

Lucasian Primality Criterion for Specific Class of 13*2^n+1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of 13*2^n+1 is introduced .
Category: Number Theory

[754] viXra:1408.0230 [pdf] submitted on 2014-08-31 12:10:44

Conjectured Compositeness Tests for Specific Classes of K10^n-C and K10^n+c

Authors: Predrag Terzic
Comments: 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k10^n-c and k10^n+c are introduced .
Category: Number Theory

[753] viXra:1408.0225 [pdf] submitted on 2014-08-31 00:12:58

Four Unusual Conjectures on Primes Involving Egyptian Fractions

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make four conjectures on primes, conjectures which involve the sums of distinct unit fractions such as 1/p(1) + 1/p(2) + (...), where p(1), p(2), (...) are distinct primes, more specifically the periods of the rational numbers which are the results of the sums mentioned above.
Category: Number Theory

[752] viXra:1408.0223 [pdf] submitted on 2014-08-31 01:36:10

Three Formulas that Generate Easily Certain Types of Triplets of Primes

Authors: Marius Coman
Comments: 2 Pages.

In this paper I present three formulas, each of them with the following property: starting from a given prime p, are obtained in many cases two other primes, q and r. I met the triplets of primes [p, q, r] obtained with these formulas in the study of Carmichael numbers; the three primes mentioned are often the three prime factors of a 3-Carmichael number.
Category: Number Theory

[751] viXra:1408.0221 [pdf] submitted on 2014-08-31 06:11:45

A New Bold Conjecture About a Way in Which Any Prime Can be Written

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make a conjecture which states that any prime greater than or equal to 53 can be written at least in one way as a sum of three odd primes, not necessarily distinct, of the same form from the following four ones: 10k + 1, 10k + 3, 10k + 7 or 10k + 9.
Category: Number Theory

[750] viXra:1408.0220 [pdf] submitted on 2014-08-31 06:41:55

A Bold Conjecture About a Way in Which Any Square of Prime Can be Written

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make a conjecture which states that any square of a prime greater than or equal to 7 can be written at least in one way as a sum of three odd primes, not necessarily distinct, but all three of the form 10k + 3 or all three of the form 10k + 7.
Category: Number Theory

[749] viXra:1408.0218 [pdf] submitted on 2014-08-30 12:33:04

Lucasian Primality Criterion for Specific Class of 7*2^n+1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of 7*2^n+1 is introduced .
Category: Number Theory

[748] viXra:1408.0217 [pdf] submitted on 2014-08-30 12:34:57

Lucasian Primality Criterion for Specific Class of 11*2^n+1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of 11*2^n+1 is introduced .
Category: Number Theory

[747] viXra:1408.0212 [pdf] submitted on 2014-08-29 14:54:03

Proof that an Infinite Number of Mersenne Prime Numbers Exit

Authors: Stephen Marshall
Comments: 11 Pages.

This paper presents a complete and exhaustive proof of the infinitude of Mersenne prime numbers. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n (see reference 1 and 2): n=(p-1)!(1/p+(-1)dd!/(p + d))+1/p+ 1/(p+d) We use this proof for d = 2p(k+m) - 2p(k) to prove the infinitude of Mersenne prime numbers.
Category: Number Theory

[746] viXra:1408.0210 [pdf] submitted on 2014-08-29 11:21:12

An Amazing Formula for Producing Big Primes Based on the Numbers 25 and 906304

Authors: Marius Coman
Comments: 3 Pages.

In this paper I present a formula for generating big primes and products of very few primes, based on the numbers 25 and 906304, formula equally extremely interesting and extremely simple, id est 25^n + 906304. This formula produces for n from 1 to 30 (and for n = 30 is obtained a number p with not less than 42 digits) only primes or products of maximum four prime factors.
Category: Number Theory

[745] viXra:1408.0209 [pdf] submitted on 2014-08-29 12:10:30

Proof That an Infinite Number of Sophie Germain Primes Exist

Authors: Stephen Marshall
Comments: 6 Pages.

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 29 is a Sophie Germain prime because it is a prime and 2 × 29 + 1 = 59, and 59 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain. We shall prove that there are an infinite number of Sophie Germain primes.
Category: Number Theory

[744] viXra:1408.0208 [pdf] submitted on 2014-08-29 07:28:09

Lucasian Primality Criterion for Specific Class of 5*2^n+1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of 5*2^n+1 is introduced .
Category: Number Theory

[743] viXra:1408.0201 [pdf] submitted on 2014-08-28 15:30:00

Proof of Infinite Number of Triplet Primes

Authors: Stephen Marshall
Comments: 12 Pages.

This paper presents a complete and exhaustive proof that an Infinite Number of Triplet Primes exist. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n (see reference 1 and 2): n =(p−1)!(1/p+(−1)d(d!)/(p + d)+ 1/(p+1)+ 1/(p+d) We use this proof and Euclid logic to prove only an infinite number of Triplet Primes exist. However we shall begin by assuming that a finite number of Triplet Primes exist, we shall prove a contradiction to the assumption of a finite number, which will prove that an infinite number of Triplet Primes exist.
Category: Number Theory

[742] viXra:1408.0197 [pdf] submitted on 2014-08-28 12:50:19

The Sequence of the Primes

Authors: Anibal Fernando Barral
Comments: 24 Pages.

In mathematics, a prime number is a natural number that is divisible only by 1 and itself. For centuries, the search for an algorithm that could generate the sequence of these numbers became a mystery. Perhaps the problem arises at the beginning of the enterprise, that is, the search for a single algorithm. I noticed that all the primes without exception increased by one unit in some cases, or decreased by one unit in the other cases result in a multiple of 6 (six) Example: 5+1=6 ; 7-1=6 ; 11+1=12 ; 13-1=12 ; 17+1=18 ; 19-1=18 ; 23+1=24 ; 29+1=30 ; 31-1=30 ; 37-1=36 ; 41+1=42 ; 43-1=42 ; 47+1=48 ; and so on. Then I thought of making it easier to split the problem solving both cases. So are passed to assume the presence of # 2 complementary families of primes. To the number 1000, I worked by hand, a job with some effort but great satisfaction. At this point my algorithms were reliable, but I needed another test. To get to number 60,000 I leaned in a computational program, which compiled a dear friend. I would have liked to get up to 1,000,000 but the limit of 60,000 has been imposed by the processing time of the data. At this point I had no more doubts about the reliability of my algorithms that are developed in continuation.
Category: Number Theory

[741] viXra:1408.0195 [pdf] submitted on 2014-08-28 08:44:01

Comments on Recent Papers by S. Marshall Claiming Proofs of Several Conjectures in Number Theory

Authors: Matthias Lesch
Comments: 3 Pages.

In recent three preprints S. Marshall claims to give proofs of several famous conjectures in number theory, among them the twin prime conjecture and Goldbach's conjecture. A claimed proof of Beal's conjecture would even imply an elementary proof of Fermat's Last Theorem. It is the purpose of this note to point out serious errors. It is the opinion of this author that it is safe to say that the claims of the above mentioned papers are lacking any basis.
Category: Number Theory

[740] viXra:1408.0193 [pdf] submitted on 2014-08-27 18:59:21

Generalized Expansions of Real Numbers

Authors: Simon Plouffe
Comments: 38 Pages.

I present here a collection of algorithms that permits the expansion into a finite series or sequence from a real number x∈ R, the precision used is 64 decimal digits. The collection of mathematical constants was taken from my own collection and theses sources [1]-[6][9][10]. The goal of this experiment is to find a closed form of the sequence generated by the algorithm. Some new results are presented.  
Category: Number Theory

[739] viXra:1408.0190 [pdf] submitted on 2014-08-27 23:33:11

An Answer to Beal's Conjecture

Authors: Francis Thasayyan
Comments: 3 Pages.

This document gives an answer to Beal's Conjection.
Category: Number Theory

[738] viXra:1408.0189 [pdf] submitted on 2014-08-28 00:37:39

Lucasian Primality Criterion for Specific Class of 9*2^n+1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of 9*2^n+1 is introduced .
Category: Number Theory

[737] viXra:1408.0184 [pdf] submitted on 2014-08-27 09:13:25

Lucasian Primality Criterion for Specific Class of K6^n-1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class of numbers of the form k6^n-1 is introduced .
Category: Number Theory

[736] viXra:1408.0183 [pdf] submitted on 2014-08-27 05:41:21

Lucasian Primality Criterion for Specific Class of Kb^n-1

Authors: Predrag Terzic
Comments: 1 Page.

Conjectured polynomial time primality test for specific class@@ of numbers of the form kb^n-1 is introduced .
Category: Number Theory

[735] viXra:1408.0181 [pdf] submitted on 2014-08-26 22:22:43

On a Strange Class of Algebraic Numbers

Authors: Simon Plouffe
Comments: 9 Pages. The abstract in english and the main text in french

The iteration formula Z_(n+1)=Z_n^2+c of Mandelbrot will give an algebraic number of degree 4 when it converges most of the time. If we take a good look at some of these algebraic numbers: some of them have a very persistent pattern in their binary expansion. La formule d’itération de Mandelbrot Z_(n+1)=Z_n^2+c converge vers un nombre algébrique de degré 4 si c est un rationnel simple. Mais en regardant de près certains nombres algébriques en binaire on voit apparaître un motif assez évident et très persistant.
Category: Number Theory

[734] viXra:1408.0180 [pdf] submitted on 2014-08-26 22:24:59

On the Values of the Function Zeta and Gamma

Authors: Simon Plouffe
Comments: 13 Pages. The abstract in english and the main text in french

An analysis of the function 1/π Arg ζ((1/2)+in) is presented. This analysis permits to find a general expression for that function using elementary functions of floor and fractional part. These formulas bring light to a remark from Freeman Dyson which relates the values of the ζfunction to quasi-crystals. We find these same values for another function which is very similar, namely 1/π Arg Γ((1/4)+in/2). These 2 sets of formula have a definite pattern, the n’th term is related to values like π,ln⁡(π),ln⁡(2),…,log⁡(p), where p is a prime number. The coefficients are closed related to a certain sequence of numbers which counts the number of 0’s from the right in the binary representation of n. These approximations are regular enough to deduce an asymptotic and precise formula. All results presented here are empirical.
Category: Number Theory

[733] viXra:1408.0176 [pdf] submitted on 2014-08-26 07:18:46

Goldbach's Conjecture. Demonstration by Analysis of Arithmetic Progressions.

Authors: Ramón Ruiz
Comments: 34 Pages. This research is based on an approach developed solely to demonstrate the binary Goldbach Conjecture and the Twin Primes Conjecture.

Goldbach's Conjecture statement: “Every even integer greater than 2 can be expressed as the sum of two primes”. Initially, to prove this conjecture, we can form two arithmetic sequences (A and B) different for each even number, with all the natural numbers that can be primes, that can added, in pairs, result in the corresponding even number. By analyzing the pairing process, in general, between all non-prime numbers of sequence A, with terms of sequence B, or vice versa, to obtain the even number, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula, to calculate the approximate number of pairs of primes that meet the conjecture for an even number x. The result of this formula is always equal or greater than 1, and it tends to infinite when x tends to infinite, which allow us to confirm that Goldbach's Conjecture is true. The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.
Category: Number Theory

[732] viXra:1408.0175 [pdf] submitted on 2014-08-26 07:27:11

Twin Primes Conjecture. Demonstration by Analysis of Arithmetic Progressions.

Authors: Ramón Ruiz
Comments: 24 Pages. This research is based on an approach developed solely to demonstrate the Twin Primes Conjecture and the binary Goldbach Conjecture.

Twin Primes Conjecture statement: “There are infinitely many primes p such that (p + 2) is also prime”. Initially, to prove this conjecture, we can form two arithmetic sequences (A and B), with all the natural numbers, lesser than a number x, that can be primes and being each term of sequence B equal to its partner of sequence A plus 2. By analyzing the pairing process, in general, between all non-prime numbers of sequence A, with terms of sequence B, or vice versa, we note that some pairs of primes are always formed. This allow us to develop a non-probabilistic formula to calculate the approximate number of pairs of primes, p and (p + 2), that are lesser than x. The result of this formula tends to infinite when x tends to infinite, which allow us to confirm that the Twin Primes Conjecture is true. The prime numbers theorem by Carl Friedrich Gauss, the prime numbers theorem in arithmetic progressions and some axioms have been used to complete this investigation.
Category: Number Theory

[731] viXra:1408.0174 [pdf] submitted on 2014-08-26 08:02:11

Proofs of Polignac Prime Conjecture, Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture

Authors: Stephen Marshall
Comments: 10 Pages.

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer n (see reference 1 and 2): n =(p−1)!(1/p+(−1)d(d!)/(p + d)+ 1/(p+1)+ 1/(p+d) We use this proof for d = 2k to prove the infinitude of Polignac prime numbers. Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.
Category: Number Theory

[730] viXra:1408.0173 [pdf] submitted on 2014-08-26 08:10:03

Proof of Beal’s Conjecture

Authors: Stephen Marshall
Comments: 7 Pages.

Abstract: This paper presents a complete and exhaustive proof of the Beal Conjecture. The approach to this proof uses the Fundamental Theorem of Arithmetic as the basis for the proof of the Beal Conjecture. The Fundamental Theorem of Arithmetic states that every number greater than 1 is either prime itself or is unique product of prime numbers. The prime factorization of every number greater than 1 is used throughout every section of the proof of the Beal Conjecture. Without the Fundamental Theorem of Arithmetic, this approach to proving the Beal Conjecture would not be possible.
Category: Number Theory

Replacements of recent Submissions

[371] viXra:1410.0120 [pdf] replaced on 2014-10-23 11:01:01

Goldbach´s Conjecture Implies Twin Prime Conjecture

Authors: O.Emilio C.Sánchez
Comments: 2 Pages.

These two conjectures are perhaps the two most famous unsolved problems in number theory but, in fact, they are closely linked. In this concise article we will prove that Goldbach´s conjecture implies twin prime´s. Whether twin prime´s conjecture implies Goldbach´s conjecture (and then both conjectures would became equivalent) or not, will be matter of further and hard working.
Category: Number Theory

[370] viXra:1410.0114 [pdf] replaced on 2014-10-21 17:30:41

The Non-existence of Odd Perfect Numbers

Authors: A. Garcés Doz
Comments: 8 Pages. Perfecting the demonstration

This proof uses a congruence, which is implicit in the condition, mandatory, demonstrated by Euler. More precisely, a congruence that must be fulfilled in the equation that equals the odd number 2N, with Euler condition, and the formula for the sum of the divisors of the number N. Following a rigorous and meticulous way, this mandatory congruence; a final equation is obtained after several polynomials simplifications on both sides of the original equation that equals the number 2N with the sum of the divisors of the number N. With this final equation, the impossibility of the existence of odd perfect numbers is demonstrated by applying several lemmas ,which lead to the impossibility of the existence of odd perfect numbers by the impossibility of satisfiability of a condition of parity and thus lead to the inexistence of odd perfect numbers \;2N=\sigma(N)\; is fulfilled.
Category: Number Theory

[369] viXra:1410.0066 [pdf] replaced on 2014-10-18 07:04:05

Notes on the Proof of Second Hardy-Littlewood Conjecture

Authors: S. Roy
Comments: 5 Pages.

In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture, namely that inequality $\pi(x)+\pi(y) > \pi (x+y)$ s examined, where $\pi(x)$ denotes the number of primes not exceeding $x$. It is shown that the inequality holds for all sufficiently large x and y. It has also been shown that for a given value of $y \geq 55$ the inequality $\pi(x)+\pi(y) > \pi (x+y)$ holds for all sufficiently large $x$. Finally, in the concluding section an argument has been given to completely settle the conjecture.
Category: Number Theory

[368] viXra:1410.0066 [pdf] replaced on 2014-10-13 07:49:37

Notes On the Proof of Second Hardy-Littlewood Conjecture

Authors: S. Roy
Comments: 4 Pages.

In this paper a slightly stronger version of the Second Hardy-Littlewood Conjecture, namely the inequality $\pi(x) + \pi(y) > \pi(x + y)$ is examined, where $\pi(x)$ denotes the number of primes not exceeding $x$. It is shown that there the inequalty holds for all suciently large $x$ and $y$.
Category: Number Theory

[367] viXra:1410.0061 [pdf] replaced on 2014-10-22 16:05:56

Relations Invariantes Entre Nombres de Décompositions de Goldbach Codées Dans un Langage à 4 Lettres

Authors: Denise Vella-Chemla
Comments: 23 Pages.

On propose une modélisation des décompositions binaires de Goldbach dans un langage à 4 lettres qui permettent de découvrir des relations invariantes entre nombres de décompositions.
Category: Number Theory

[366] viXra:1410.0054 [pdf] replaced on 2014-10-23 19:23:16

The Art of Inspired Guessing

Authors: Simon Plouffe
Comments: 6 Pages. minor corrections

A presentation of various formulas is given. Many of these findings have no explanation whatsoever. One is related to the mass ratio of the neutron and proton: 1.00137841917. Other expression are given to the mass ratio of the neutron and the electron. They were found using a variety of methods using either a HP‐15C calculator in 1988 to the current database of constants of the author which consist of 13.155 billion entries.
Category: Number Theory

[365] viXra:1410.0054 [pdf] replaced on 2014-10-19 01:42:06

The Art of Inspired Guessing

Authors: Simon Plouffe
Comments: 6 Pages. new formulas for mass ratios

A presentation of various formulas is given. Many of these findings have no explanation whatsoever. One is related to the mass ratio of the neutron and proton: 1.00137841917. Other expression are given to the mass ratio of the neutron and the electron. They were found using a variety of methods using either a HP‐15C calculator in 1988 to the current database of constants of the author which consist of 13.155 billion entries.
Category: Number Theory

[364] viXra:1410.0054 [pdf] replaced on 2014-10-18 18:42:23

The Art of Inspired Guessing

Authors: Simon Plouffe
Comments: 6 Pages. an additional formula for the mass ratio of the proton and electron : 1/5/sinh(Pi)+6*Pi^5+1/5/cosh(Pi)

A presentation of various formulas is given. Many of these findings have no explanation whatsoever. One is related to the mass ratio of the neutron and proton: 1.00137841917. They were found using a variety of methods using either a HP‐15C calculator in 1988 to the current database of constants of the author which consist of 12.3 billion entries.
Category: Number Theory

[363] viXra:1410.0038 [pdf] replaced on 2014-10-11 10:08:27

A Modification of Riesel Primality Test

Authors: Predrag Terzic
Comments: 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form k2^n-1 is introduced .
Category: Number Theory

[362] viXra:1410.0030 [pdf] replaced on 2014-10-13 07:43:15

The Generalized Goldbach's Conjecture: the Corollary of Symmetry Primes

Authors: Jian Ye
Comments: 3 Pages.

Goldbach’s conjecture: symmetry primes exists in natural numbers. the generalized Goldbach’s conjecture: symmetry of primes in the former and tolerance coprime to arithmetic progression still exists.
Category: Number Theory

[361] viXra:1409.0155 [pdf] replaced on 2014-10-10 21:42:19

The Sum of the Digits of a Number/primality Testing

Authors: Ounas Meriam
Comments: 04 Pages.

In this paper, I will try to explain my idea about the world of the digits of numbers which is somewhat circumvented by mathematicians.
Category: Number Theory

[360] viXra:1409.0100 [pdf] replaced on 2014-09-13 10:46:52

The Lattice Reduction Algorithm and Applications (LLL and PSLQ)

Authors: Simon Plouffe
Comments: 29 Pages. Conference in Montréal and Vancouver 1995-1996

A survey of Integer Relations algorithms such as LLL or PSLQ, some examples are given. A method to get the algebraic generating function from a finite series is given.
Category: Number Theory

[359] viXra:1409.0095 [pdf] replaced on 2014-09-12 22:03:23

Computing the Generating Function of a Series Given Its First Few Terms

Authors: Simon Plouffe, François Bergeron
Comments: 6 Pages.

We outline an approach for the computation of a good can- didate for the generating function of a power series for which only the first few coefficients are known. More precisely, if the derivative, the logarithmic derivative, the reversion, or another transformation of a given power series (even with polynomial coefficients) appears to admit a rational generating function, we compute the generating function of the original series by applying the inverse of those transformations to the rational generating function found.
Category: Number Theory

[358] viXra:1409.0093 [pdf] replaced on 2014-09-13 10:50:09

On the Rapid Computation of Various Polylogarithmic Constants

Authors: Simon Plouffe, David Bailey, Peter Borwein
Comments: 13 Pages. a better copy

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or π on a modest work station in a few hours run time.
Category: Number Theory

[357] viXra:1409.0052 [pdf] replaced on 2014-09-14 17:15:02

When π(N) Does not Divide N

Authors: Germán Paz
Comments: 6 Pages. Main text in English; abstract in English and Spanish; title and abstract changed; some results added. /// Texto principal en inglés; resumen en inglés y en español; título y resumen modificados; algunos resultados agregados.

Let $\pi(n)$ denote the prime-counting function. In this paper we work with explicit formulas for $\pi(n)$ that are valid for infinitely many positive integers $n$, and we prove that if $n\ge 60184$ and $\ln n-\lfloor\ln n\rfloor>0.1$, then $\pi(n)$ does not divide $n$. Based on this result, we show that if $e$ is the base of the natural logarithm, $a$ is a fixed integer $\ge 11$ and $n$ is any integer in the interval $[e^{a+0.1},e^{a+1}]$, then $\pi(n)\nmid n$. In addition, we prove that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $n$ is a multiple of $\lfloor\ln n-1\rfloor$ located in the interval $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.

///////////////////

Sea $\pi(n)$ la función contadora de números primos. En este documento trabajamos con funciones explícitas para $\pi(n)$ que son válidas para infinitos enteros positivos $n$, y demostramos que si $n\ge 60184$ y $\ln n-\lfloor\ln n\rfloor>0.1$, entonces $\pi(n)$ no divide a $n$. Basándonos en este resultado, probamos que si $e$ es la base del logaritmo natural, $a$ un entero fijo $\ge 11$ y $n$ un entero cualquiera dentro del intervalo $[e^{a+0.1},e^{a+1}]$, entonces $\pi(n)\nmid n$. Además, demostramos que si $n\ge 60184$ y $\pi(n)$ divide a $n$, entonces $n$ es un múltiplo de $\lfloor\ln n-1\rfloor$ ubicado en el intervalo $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.
Category: Number Theory

[356] viXra:1409.0052 [pdf] replaced on 2014-09-13 23:58:49

When π(N) Does not Divide N

Authors: Germán Paz
Comments: 6 Pages. Main text in English; abstract in English and Spanish; title and abstract changed; some results added. /// Texto principal en inglés; resumen en inglés y en español; título y resumen modificados; algunos resultados agregados.

Let $\pi(n)$ denote the prime-counting function. In this paper we work with explicit formulas for $\pi(n)$ that are valid for infinitely many positive integers $n$, and we prove that if $n\ge 60184$ and $\ln n-\lfloor\ln n\rfloor>0.1$, then $\pi(n)$ does not divide $n$. Based on this result, we show that if $e$ is the base of the natural logarithm, $a$ is a fixed integer $\ge 11$ and $n$ is any integer in the interval $[e^{a+0.1},e^{a+1}]$, then $\pi(n)\nmid n$. In addition, we prove that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $n$ is a multiple of $\lfloor\ln n-1\rfloor$ located in the interval $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.

///////////////////

Sea $\pi(n)$ la función contadora de números primos. En este documento trabajamos con funciones explícitas para $\pi(n)$ que son válidas para infinitos enteros positivos $n$, y demostramos que si $n\ge 60184$ y $\ln n-\lfloor\ln n\rfloor>0.1$, entonces $\pi(n)$ no divide a $n$. Basándonos en este resultado, probamos que si $e$ es la base del logaritmo natural, $a$ un entero fijo $\ge 11$ y $n$ un entero cualquiera dentro del intervalo $[e^{a+0.1},e^{a+1}]$, entonces $\pi(n)\nmid n$. Además, demostramos que si $n\ge 60184$ y $\pi(n)$ divide a $n$, entonces $n$ es un múltiplo de $\lfloor\ln n-1\rfloor$ ubicado en el intervalo $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.
Category: Number Theory

[355] viXra:1409.0052 [pdf] replaced on 2014-09-13 00:15:41

When π(N) Does not Divide N

Authors: Germán Paz
Comments: 6 Pages. Main text in English; abstract in English and Spanish; title and abstract changed; some results added. /// Texto principal en inglés; resumen en inglés y en español; título y resumen modificados; algunos resultados agregados.

Let $\pi(n)$ denote the prime-counting function. In this paper we work with explicit formulas for $\pi(n)$ that are valid for infinitely many positive integers $n$, and we prove that if $n\ge 60184$ and $\ln n-\lfloor\ln n\rfloor>0.1$, then $\pi(n)$ does not divide $n$. Based on this result, we show that if $e$ is the base of the natural logarithm, $a$ is a fixed integer $\ge 11$ and $n$ is any integer in the interval $[e^{a+0.1},e^{a+1}]$, then $\pi(n)\nmid n$. In addition, we prove that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $n$ is a multiple of $\lfloor\ln n-1\rfloor$ located in the interval $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.

///////////////////

Sea $\pi(n)$ la función contadora de números primos. En este documento trabajamos con funciones explícitas para $\pi(n)$ que son válidas para infinitos enteros positivos $n$, y demostramos que si $n\ge 60184$ y $\ln n-\lfloor\ln n\rfloor>0.1$, entonces $\pi(n)$ no divide a $n$. Basándonos en este resultado, probamos que si $e$ es la base del logaritmo natural, $a$ un entero fijo $\ge 11$ y $n$ un entero cualquiera dentro del intervalo $[e^{a+0.1},e^{a+1}]$, entonces $\pi(n)\nmid n$. Además, demostramos que si $n\ge 60184$ y $\pi(n)$ divide a $n$, entonces $n$ es un múltiplo de $\lfloor\ln n-1\rfloor$ ubicado en el intervalo $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.
Category: Number Theory

[354] viXra:1409.0052 [pdf] replaced on 2014-09-10 03:14:15

When π(N) Divides N and When it Does not

Authors: Germán Paz
Comments: 7 Pages. Main text in English; abstract in English and Spanish. New results added in version 2. /// Texto principal en inglés; resumen en inglés y en español. Nuevos resultados agregados en la versión 2.

Let $\pi(n)$ denote the prime-counting function. In this paper we work with explicit formulas for $\pi(n)$ that are valid for infinitely many positive integers $n$, and we prove that if $n\ge 60184$ and $\operatorname{frac}(\ln n)=\ln n-\lfloor\ln n\rfloor>0.5$, then $\pi(n)$ does not divide $n$. Based on this result, we show that if $e$ is the base of the natural logarithm, $a$ is a fixed integer $\ge 11$ and $n$ is any integer in the interval $[e^{a+0.5},e^{a+1}]$, then $\pi(n)\nmid n$. In addition, we prove that if $n\ge 60184$ and $n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\ln n-1\rfloor$ located in the interval $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.

///////////////////

Sea $\pi(n)$ la función contadora de números primos. En este documento trabajamos con funciones explícitas para $\pi(n)$ que son válidas para infinitos enteros positivos $n$, y demostramos que si $n\ge 60184$ y $\operatorname{frac}(\ln n)=\ln n-\lfloor\ln n\rfloor>0.5$, entonces $\pi(n)$ no divide a $n$. Basándonos en este resultado, probamos que si $e$ es la base del logaritmo natural, $a$ es un entero fijo $\ge 11$ y $n$ es cualquier entero en el intervalo $[e^{a+0.5},e^{a+1}]$, entonces $\pi(n)\nmid n$. Además, demostramos que si $n\ge 60184$ y $n/\pi(n)$ es entero, entonces $n$ es un múltiplo de $\lfloor\ln n-1\rfloor$ ubicado en el intervalo $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.
Category: Number Theory

[353] viXra:1409.0039 [pdf] replaced on 2014-10-10 03:58:08

Over One Hundred and Fifty Conjectures on Primes, Many Based on the Observation of Fermat Pseudoprimes

Authors: Marius Coman
Comments: 92 Pages.

In two of my previous published books, “Two hundred conjectures and one hundred and fifty open problems on primes”, respectively “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, I already expressed my conviction that the study of Fermat pseudoprimes, fascinating numbers that seem to be a little bit more willing to let themselves ordered and understood than the prime numbers, can help a lot in understanding these latter. This book brings together forty papers on prime numbers, many of them supporting the author’s belief, expressed above, namely that new ordered patterns can be discovered in the “undisciplined” set of prime numbers, observing the ordered patterns in the set of Fermat pseudoprimes, especially in the set of Carmichael numbers, the absolute Fermat pseudoprimes, and in the set of Poulet (sometimes also called Sarrus) numbers, the relative Fermat pseudoprimes to base two. Few papers, which are not based on the observation of pseudoprimes, though apparently heterogenous, still have something in common: they are all directed toward the same goal, discovery of new patterns in the set of primes, using the same means, namely the old and reliable integers. Part One of this book of collected papers contains over one hundred and fifty conjectures on primes and Part Two of this book brings together the articles regarding primes, submitted by the author to the preprint scientific database Vixra, representing the context of the conjectures listed in Part One.
Category: Number Theory

[352] viXra:1409.0034 [pdf] replaced on 2014-09-06 05:25:15

Four Conjectures Based on the Observation of a Type of Recurrent Sequences Involving Semiprimes

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make four conjectures starting from the observation of the following recurrent relations: (((p*q – p)*2 – p)*2 – p)...), respectively (((p*q – q)*2 – q)*2 – q)...), where p, q are distinct odd primes.
Category: Number Theory

[351] viXra:1408.0195 [pdf] replaced on 2014-09-13 01:16:26

Comments on Recent Papers by S. Marshall Claiming Proofs of Several Conjectures in Number Theory

Authors: Matthias Lesch
Comments: 3 Pages.

In a recent series of preprints S. Marshall claims to give proofs of several famous conjectures in number theory, among them the twin prime conjecture and Goldbach’s conjecture. A claimed proof of Beal’s conjecture would even imply an elemen- tary proof of Fermat’s Last Theorem. It is the purpose of this note to point out serious errors. It is the opinion of this author that it is safe to say that the claims of the above mentioned papers are lacking any basis.
Category: Number Theory

[350] viXra:1408.0174 [pdf] replaced on 2014-10-07 09:10:22

Proofs of Polignac Prime Conjecture, Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture

Authors: Stephen Marshall
Comments: 10 Pages. This is an updated proof by the author.

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n: n = (p−1)!(1/p+(−1)d(d!)/(p + d)+ 1/(p+1)+ 1/(p+d) We use this proof for d = 2k to prove the infinitude of Polignac prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Polignac Prime Conjecture possible. Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.
Category: Number Theory