**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(3)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)

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(18)

Any replacements are listed further down

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

**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

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

**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

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

**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

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

**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

[815] **viXra:1410.0090 [pdf]**
*submitted on 2014-10-16 15:19:26*

**Authors:** O.Emilio.C.Sánchez

**Comments:** 4 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 both conjectures are equivalent . First we prove Twin prime conjecture implies Goldbach´s conjecture; second, we prove Goldbach´s conjecture implies twin prime´s.

**Category:** Number Theory

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.[766] **viXra:1409.0048 [pdf]**
*submitted on 2014-09-07 11:22:05*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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

[729] **viXra:1408.0169 [pdf]**
*submitted on 2014-08-25 18:53:30*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

This paper presents a complete and exhaustive proof of the Fibonacci 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 p = Fy-1 and d = Fy-2 to prove the infinitude of Fibonacci prime numbers.

**Category:** Number Theory

[728] **viXra:1408.0166 [pdf]**
*submitted on 2014-08-25 09:29:06*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 3*2^n+1 is introduced .

**Category:** Number Theory

[727] **viXra:1408.0150 [pdf]**
*submitted on 2014-08-23 02:35:15*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt does not require knowledge of the distribution of primes.

**Category:** Number Theory

[726] **viXra:1408.0134 [pdf]**
*submitted on 2014-08-20 08:04:44*

**Authors:** Predrag Terzic

**Comments:** 4 Pages.

Conjectured polynomial time primality and compositeness tests for numbers of special forms are introduced .

**Category:** Number Theory

[725] **viXra:1408.0128 [pdf]**
*submitted on 2014-08-19 05:07:11*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

We use the positivity axiom of inner product spaces to prove the equivalent statement of the Riemann hypothesis.

**Category:** Number Theory

[724] **viXra:1408.0126 [pdf]**
*submitted on 2014-08-18 15:16:53*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality tests for specific classes of numbers of the form kb^n-1 are introduced .

**Category:** Number Theory

[723] **viXra:1408.0119 [pdf]**
*submitted on 2014-08-18 09:49:52*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form 9b^n-1 is introduced .

**Category:** Number Theory

[722] **viXra:1408.0113 [pdf]**
*submitted on 2014-08-18 06:11:05*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make five conjectures about the primes r, t and the square of prime p^2, which appears as solutions in the diophantine equation 120*n*q*r + 1 = p^2, where n is non-null positive integer.

**Category:** Number Theory

[721] **viXra:1408.0111 [pdf]**
*submitted on 2014-08-18 02:11:31*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures abut the pairs of primes [p1, q1], where the difference between p1 and q1 is a certain even number d. I state that any such pair has at least one other corresponding, in a specified manner, pair of primes [p2, q2], such that the difference between p2 and q2 is also equal to d.

**Category:** Number Theory

[720] **viXra:1408.0110 [pdf]**
*submitted on 2014-08-18 00:02:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make a conjecture which states that any odd prime can be written in a certain way, in other words that any such prime can be expressed using just another prime and the powers of the numbers 2, 3 and 5. I also make a related conjecture about twin primes.

**Category:** Number Theory

[719] **viXra:1408.0098 [pdf]**
*submitted on 2014-08-16 08:37:00*

**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

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

**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

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

**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

[361] **viXra:1410.0061 [pdf]**
*replaced on 2014-10-18 14:41:35*

**Authors:** Denise Vella-Chemla

**Comments:** 22 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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[351] **viXra:1409.0052 [pdf]**
*replaced on 2014-09-13 23:58:49*

**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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[350] **viXra:1409.0052 [pdf]**
*replaced on 2014-09-13 00:15:41*

**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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[349] **viXra:1409.0052 [pdf]**
*replaced on 2014-09-10 03:14:15*

**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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.[348] **viXra:1409.0039 [pdf]**
*replaced on 2014-10-10 03:58:08*

**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

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

**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

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

**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

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

**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

[344] **viXra:1408.0166 [pdf]**
*replaced on 2014-08-27 05:29:01*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 3*2^n+1 is introduced .

**Category:** Number Theory

[343] **viXra:1408.0166 [pdf]**
*replaced on 2014-08-25 12:25:49*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of 3*2^n+1 is introduced .

**Category:** Number Theory

[342] **viXra:1408.0150 [pdf]**
*replaced on 2014-09-25 04:49:05*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This (quasi-inductive) attempt does not require knowledge of a distribution of primes.

**Category:** Number Theory

[341] **viXra:1408.0150 [pdf]**
*replaced on 2014-09-18 11:52:13*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt uses induction and therefore does not require knowledge of a distribution of primes.

**Category:** Number Theory

[340] **viXra:1408.0150 [pdf]**
*replaced on 2014-08-25 12:50:35*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt does not require knowledge of the distribution of primes.

**Category:** Number Theory

[339] **viXra:1408.0134 [pdf]**
*replaced on 2014-09-18 08:45:00*

**Authors:** Predrag Terzic

**Comments:** 7 Pages.

Conjectured polynomial time primality and compositeness tests for numbers of special forms are introduced .

**Category:** Number Theory

[338] **viXra:1408.0134 [pdf]**
*replaced on 2014-08-27 05:27:43*

**Authors:** Predrag Terzic

**Comments:** 4 Pages.

Conjectured polynomial time primality and compositeness tests for numbers of special forms are introduced .

**Category:** Number Theory

[337] **viXra:1408.0126 [pdf]**
*replaced on 2014-08-27 05:23:44*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality tests for specific classes of numbers of the form kb^n-1 are introduced .

**Category:** Number Theory

[336] **viXra:1408.0113 [pdf]**
*replaced on 2014-08-18 06:42:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make five conjectures about the primes q, r and the square of prime p^2, which appears as solutions in the diophantine equation 120*n*q*r + 1 = p^2, where n is non-null positive integer.

**Category:** Number Theory