**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(12) - 1405(17) - 1406(21) - 1407(35) - 1408(52) - 1409(41)

Any replacements are listed further down

[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

[718] **viXra:1408.0095 [pdf]**
*submitted on 2014-08-16 05:39:21*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time primality test for specific class of numbers of the form 3b^n-1 is introduced .

**Category:** Number Theory

[717] **viXra:1408.0087 [pdf]**
*submitted on 2014-08-14 07:34:55*

**Authors:** William Maclachlan

**Comments:** 11 Pages.

The aim of my "experiment" was to gather some curious information about the understanding of primes- to my understanding I seemed to have created a system that can find primes considerably quicker in contrast to merely searching through all the given number's factors.
I am not a professional, but it would be nice if I could get some form of a reply from someone with experience to explain the irrelevancy of my findings.

**Category:** Number Theory

[716] **viXra:1408.0085 [pdf]**
*submitted on 2014-08-14 03:16:30*

**Authors:** Pingyuan Zhou

**Comments:** 5 Pages. Author gives an argument for the infinity of primes of the form 2x^2-1 by the infinity of near-square primes of Mersenne primes to arise from a corresponding Fermat prime criterion.

Abstract: In this paper we consider primes of the form 2x^2-1 and discover there is a very great probability for appearing of such primes, and give an argument for the infinity of primes of the form 2x^2-1 by the infinity of near-square primes of Mersenne primes.

**Category:** Number Theory

[715] **viXra:1408.0083 [pdf]**
*submitted on 2014-08-14 00:17:08*

**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

[714] **viXra:1408.0079 [pdf]**
*submitted on 2014-08-13 07:26:37*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k2^n-c and k2^n+c are introduced .

**Category:** Number Theory

[713] **viXra:1408.0071 [pdf]**
*submitted on 2014-08-12 02:52:56*

**Authors:** Predrag Terzic

**Comments:** 1 Page.

Conjectured polynomial time primality test for specific class of generalized Fermat numbers is introduced .

**Category:** Number Theory

[712] **viXra:1408.0068 [pdf]**
*submitted on 2014-08-11 10:15:02*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k2^n-1 and k2^n+1 are introduced .

**Category:** Number Theory

[711] **viXra:1408.0050 [pdf]**
*submitted on 2014-08-08 18:22:49*

**Authors:** Oh Jung Uk

**Comments:** 20 Pages. I don't know how to show abstract well

If π_g (N) is the number of cases that even number N could be expressed as the sum of the two primes of 6n±1 type then the formula of π_g (N) is below
π_g (6n+0)=n-1- 2/3 ∑_(k=1)^(n-1)▒((〖πβ〗_g (6k-1))/(πβ_g (6k-1)-1)) -2/3π ∑_(k=1)^(n-1)▒∑_(m=1)^∞▒sin((2〖mπ〗^2 β_g (6k-1))/(πβ_g (6k-1)-1))/m
where,β_g (6k-1)=τ(6k-1)-2+τ(6(n-k)+1)-2,…
But,the formula of π_g (6n+2),π_g (6n-2) is omitted in abstract.

**Category:** Number Theory

[710] **viXra:1408.0046 [pdf]**
*submitted on 2014-08-08 08:35:19*

**Authors:** Th. Guyer

**Comments:** 1 Page.

A briefly olympic idea about P = NP
(include the Prime_Twin_Conjecture)
Whoever is able to(o) kicks out m(e?

**Category:** Number Theory

[709] **viXra:1408.0044 [pdf]**
*submitted on 2014-08-08 04:07:06*

**Authors:** Oh Jung Uk

**Comments:** 21 Pages. I don't know how to show abstract well

If π_t (6n+1) is the number of twin prime of 6n+1 or less then the formula of π_t (6n+1) is described below.
π_t (6n+1)=n+1-2/3 ∑_(k=1)^n▒((πβ_t (6k))/(πβ_t (6k)-1)) -2/3π ∑_(k=1)^n▒∑_(m=1)^∞▒sin((2〖mπ〗^2 β_t (6k))/(πβ_t (6k)-1))/m
where,β_t (6k)={τ(6k-1)-2}+{τ(6k+1)-2},…

**Category:** Number Theory

[708] **viXra:1408.0043 [pdf]**
*submitted on 2014-08-08 04:11:24*

**Authors:** Oh Jung Uk

**Comments:** 16 Pages. I don't know how to show abstract well

For Mersenne prime of 2^(6n+1)-1 type, if a Mersenne prime is 2^(6p+1)-1, just next Mersenne prime is 2^(6x+1)-1 then the following equation is satisfied.
x =p+3/2+1/2 ∑_(k=p+1)^x▒〖(πβ(2^(6k+1)-1)+1)/(πβ(2^(6k+1)-1)-1)+1/π ∑_(k=p+1)^x▒∑_(m=1)^∞▒sin((2mπ^2 β(2^(6k+1)-1))/(πβ(2^(6k+1)-1)-1))/m〗
where,β(2^(6k+1)-1)=τ(2^(6k+1)-1)-2,…
Mersenne prime of 2^(6n-1)-1 type is omitted in abstract.

**Category:** Number Theory

[707] **viXra:1408.0042 [pdf]**
*submitted on 2014-08-08 04:16:18*

**Authors:** Oh Jung Uk

**Comments:** 12 Pages. I don't know how to show abstract well

A number of 6n-1 type is not odd perfect number, Fermat number is not also odd perfect number.
And, if Fermat number is composite number then Fermat number is factorized as below
when n is odd number,2^(2^n )+1=(2^(n+1) (3k+1)+1)(2^(n+1) (3m)+1)
when n is even number,2^(2^n )+1=(2^(n+1) ((3k+1)/2)+1)(2^(n+1) (3m)+1)
And, all Fermat number for n≥5 is composite number.

**Category:** Number Theory

[706] **viXra:1408.0041 [pdf]**
*submitted on 2014-08-07 22:23:12*

**Authors:** Oh Jung Uk

**Comments:** 34 Pages. I don't know how I can fix the abstract

The formula of prime-counting function π(N=6n+3) is described below.
π(N=6n+3)=2n+2-2/3 ∑_(k=1)^n▒{πβ(6k-1)/(πβ(6k-1)-1)+πβ(6k+1)/(πβ(6k+1)-1)} -2/3π ∑_(k=1)^n▒∑_(m=1)^∞▒{(sin((2mπ^2 β(6k-1))/(πβ(6k-1)-1))+sin((2mπ^2 β(6k+1))/(πβ(6k+1)-1)))/m}
where,β(6k-1)=τ(6k-1)-2,β(6k+1)=τ(6k+1)-2,…

**Category:** Number Theory

[705] **viXra:1408.0003 [pdf]**
*submitted on 2014-08-02 01:48:13*

**Authors:** Russell Letkeman

**Comments:** 4 Pages.

We study the spacings of numbers co-prime to an even consecutive product of primes, P_m\# and its structure exposed by the fundamental theorem of prime sieving (FTPS). We extend this to prove some parts of the Hardy-Littlewood general prime density conjecture for all finite multiplicative groups modulo a primorial. We then use the FTPS to prove such groups have gap spacings which form arithmetic progressions as long as we wish. We also establish their densities and provide prescriptions to find them.

**Category:** Number Theory

[704] **viXra:1408.0001 [pdf]**
*submitted on 2014-08-01 05:16:54*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make two conjectures, one about how could be expressed a prime of the form 6k + 1 and one about how could be expressed a prime of the form 6k – 1.

**Category:** Number Theory

[703] **viXra:1407.0224 [pdf]**
*submitted on 2014-07-30 20:04:10*

**Authors:** Russell Letkeman

**Comments:** 3 Pages.

We build a simple recursive model for the prime numbers which at its heart is the prime sieve of Eratosthenes. We also show for prime numbers greater than 3 and their gaps posses a handedness which forbids a large range of possibilities for the choice of intervals in arithmetic progressions.

**Category:** Number Theory

[702] **viXra:1407.0214 [pdf]**
*submitted on 2014-07-29 23:15:32*

**Authors:** Russell Letkeman

**Comments:** 7 Pages.

We show every set modulo the product of a random collection of unique prime numbers has a palindrome in its gaps of length the minimum set minus 1. There is one more gap which is always a twin. Together the count of the gaps equals the count of the minimum modular set. This symmetry not only forces all constellations of gaps to have mirror images existing at exactly the same counts, but it also precisely identifies the center of mass (counts) of the set.

**Category:** Number Theory

[701] **viXra:1407.0209 [pdf]**
*submitted on 2014-07-29 02:38:15*

**Authors:** Pingyuan Zhou

**Comments:** 9 Pages. Author gives an argument for indirect connections between Fermat primes and regular 2^k-sided polygons to make Gauss-Wantzel theorem have general sense in implying connections between Fermat primes and all constructible polygons.

Abstract: Gauss-Wantzel theorem shows that regular n-sided polygons, whose number of sides contains a(distrinct) Fermat prime(s) as odd prime factor(s) of n or number of sides is power of 2, are all constructible with compass and straightedge. But of these caces, the constructibility of all regular 2^k-sided polygons is not related to Fermat primes. We discover the number of so-called root Mersenne primes Mp for p

**Category:** Number Theory

[700] **viXra:1407.0205 [pdf]**
*submitted on 2014-07-27 17:21:20*

**Authors:** JinHua Fei

**Comments:** 7 Pages.

In this paper, we assume that weaker Hardy-Littlewood Conjecture, we got a better upper bound of the exceptional real zero for a class of prime number module.

**Category:** Number Theory

[699] **viXra:1407.0203 [pdf]**
*submitted on 2014-07-27 20:37:59*

**Authors:** Réjean Labrie

**Comments:** 7 Pages.

This article is a demonstration of the existence of at least one prime number between two consecutive squares.

**Category:** Number Theory

[698] **viXra:1407.0201 [pdf]**
*submitted on 2014-07-27 03:06:26*

**Authors:** T.Nakashima

**Comments:** 1 Page.

This Paper is the result Counting the Prime Numbers by using Mathematica 9.

**Category:** Number Theory

[697] **viXra:1407.0166 [pdf]**
*submitted on 2014-07-21 18:58:13*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 7 Pages.

The critical line lies on a surface. And the critical line inherits the characteristics from this surface. Then, the location of the critical line can be determined.

**Category:** Number Theory

[696] **viXra:1407.0164 [pdf]**
*submitted on 2014-07-22 01:26:41*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present two possible infinite sequences of primes, having in common the fact that their formulas contain the powers of the number 2.

**Category:** Number Theory

[695] **viXra:1407.0159 [pdf]**
*submitted on 2014-07-21 04:04:27*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make a conjecture which states that any prime greater than or equal to 5 can be written in a certain way, in other words that any such prime can be expressed using just two other primes and a power of the number 2.

**Category:** Number Theory

[694] **viXra:1407.0158 [pdf]**
*submitted on 2014-07-21 04:47:52*

**Authors:** Marius Coman

**Comments:** 3 Pages.

These conjectures state that any prime p greater than 60 can be written as a sum of three primes of a certain type from the following four ones: 10k + 1, 10k + 3, 10k + 7 and 10k + 9.

**Category:** Number Theory

[693] **viXra:1407.0157 [pdf]**
*submitted on 2014-07-21 05:43:31*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I present two possible infinite sequences of primes, having in common the fact that their formulas contain the number 360.

**Category:** Number Theory

[692] **viXra:1407.0153 [pdf]**
*submitted on 2014-07-20 23:20:01*

**Authors:** Pingyuan Zhou

**Comments:** 14 Pages. Author presents the strong finiteness of double Mersenne primes and the infinity of root Mersenne primes and near-square primes of Mersenne primes by generalizing conjecture about primality of Mersenne number.

Abstract: In this paper we present the strong finiteness of double Mersenne primes to be a subset of Mersenne primes, the infinity of so-called root Mersenne primes to be also a subset of Mersenne primes and the infinity of so-called near-square primes of Mersenne primes by generalizing our previous conjecture about primality of Mersenne number. These results and our previous results about the strong finiteness of Fermat, double Fermat and Catalan-type Fermat primes [1] give an elementary but complete understanding for the infinity or the strong finiteness of some prime number sequences of the form 2^x±1, which all have a corresponding original continuous natural ( prime ) number sequence. It is interesting that the generalization to near-square primes of Mersenne primes Wp=2(Mp)^2-1 has brought us positive result.

**Category:** Number Theory

[691] **viXra:1407.0152 [pdf]**
*submitted on 2014-07-21 02:26:52*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper are stated ten conjectures on primes, more precisely on the infinity of some types of triplets and quadruplets of primes, all of them using the multiples of the number 30 and also all of them met on the study of Carmichael numbers.

**Category:** Number Theory

[690] **viXra:1407.0151 [pdf]**
*submitted on 2014-07-21 02:50:07*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Prime number sieve using LCM function is introduced .

**Category:** Number Theory

[689] **viXra:1407.0150 [pdf]**
*submitted on 2014-07-21 03:00:29*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper are stated six conjectures on primes, more precisely on the infinity of some types of pairs of primes, all of them met in the study of 3-Carmichael numbers.

**Category:** Number Theory

[688] **viXra:1407.0143 [pdf]**
*submitted on 2014-07-19 16:20:26*

**Authors:** Isaac Mor

**Comments:** 3 Pages.

Odd Perfect Number = 36k+9
In 1953, Jacques Touchard proved that an odd perfect number must be of the form 12k + 1 or 36k + 9.
(Judy A. Holdener discovered a simpler proof of the theorem of Touchard in 2002)
if I am right then I (isaac mor lol) just showed that an odd perfect number must be of the form 36k+9 (19 july 2014)

**Category:** Number Theory

[687] **viXra:1407.0129 [pdf]**
*submitted on 2014-07-17 21:54:39*

**Authors:** Pingyuan Zhou

**Comments:** 9 Pages. In this paper, author presents the strong finiteness of Fermat primes, double Fermat primes and Catalan-type Fermat primes by generalizing previous conjecture about primality of Fermat numbers to double Fermat and Catalan-type Fermat numbers.

Abstract: In this paper we present that so-called double Fermat numbers are an infinite subset of well-known Fermat numbers and so-called Catalan-type Fermat numbers are also an infinite subset of Fermat numbers as well as double Fermat primes and Catalan-type Fermat primes are all strongly finite as Fermat primes do. From it we get the same result that composite Fermat numbers, composite double Fermat numbers and composite Catalan-type Fermat numbers are all infinite.

**Category:** Number Theory

[686] **viXra:1407.0128 [pdf]**
*submitted on 2014-07-17 13:03:45*

**Authors:** Yilun Shang

**Comments:** 5 Pages.

In this note, we consider some generalizations of the Lucas
sequence, which essentially extend sequences to triangular arrays.
Some new and elegant results are derived.

**Category:** Number Theory

[685] **viXra:1407.0117 [pdf]**
*submitted on 2014-07-15 22:13:12*

**Authors:** Pingyuan Zhou

**Comments:** 4 Pages. Aothor presents a near-sguare number sequence of all Mersenne primes, which seems to be an accptable awy in searching for larger primes by known Mersenne primes themselves than the largest known Mersenne prime M57885161.

Abstract: In this paper we present a conjecture that there is a near-square prime number sequence of Mersenne primes to arise from the near-square number sequence Wp=2(Mp)^2-1 generated from all Mersenne primes Mp, in which every term is larger prime number than corresponding perfect number. The conjecture has been verified for the first few prime terms in the near-square prime number sequence and we may expect appearing of near-square prime numbers of some known Mersenne primes with large p-values will become larger primes to be searched than the largest known Mersenne prime M57885161.

**Category:** Number Theory

[684] **viXra:1407.0111 [pdf]**
*submitted on 2014-07-15 06:26:38*

**Authors:** Choe Ryong Gil

**Comments:** 8 pages, two tables

In this paper we introduce a new function, which would be called a sigma-index of the natural
number, and consider its boundedness. This estimate is effective for the Robin inequality.

**Category:** Number Theory

[339] **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

[338] **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

[337] **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

[336] **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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[335] **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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[334] **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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.1}]$.[333] **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 $[e^{\lfloor\ln n-1\rfloor+1},e^{\lfloor\ln n-1\rfloor+1.5}]$.[332] **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

[331] **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

[330] **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

[329] **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

[328] **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

[327] **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

[326] **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

[325] **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

[324] **viXra:1408.0068 [pdf]**
*replaced on 2014-08-12 06:36:01*

**Authors:** Predrag Terzic

**Comments:** 2 Pages.

Conjectured polynomial time compositeness tests for numbers of the form k2^n-1 and k2^n+1 are introduced .

**Category:** Number Theory

[323] **viXra:1407.0214 [pdf]**
*replaced on 2014-07-31 22:10:45*

**Authors:** Russell Letkeman

**Comments:** 6 Pages.

We introduce a fundamental theorem of prime sieving (FTPS) and show how it illuminates structure on numbers co-prime to a random product of unique prime numbers. This theorem operates on the transition between the set of numbers co-prime to any product of unique prime numbers and the new set when another prime number is introduced in the product.

**Category:** Number Theory

[322] **viXra:1407.0166 [pdf]**
*replaced on 2014-07-27 13:08:59*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 7 Pages.

The critical line lies on a surface. And the critical line inherits the characteristics from this surface. Then, the location of the critical line can be determined.

**Category:** Number Theory