[47] **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

[46] **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

[45] **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

[44] **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

[43] **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

[42] **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

[41] **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

[40] **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

[39] **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

[38] **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

[37] **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

[36] **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

[35] **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

[34] **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

[33] **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

[32] **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

[31] **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

[30] **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

[29] **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

[28] **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

[27] **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

[26] **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

[25] **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

[24] **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

[23] **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

[22] **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

[21] **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

[20] **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

[19] **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

[18] **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

[17] **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

[16] **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

[15] **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

[14] **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

[13] **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

[12] **viXra:1409.0052 [pdf]**
*replaced on 2015-01-25 23:17:00*

**Authors:** Germán Paz

**Comments:** 10 Pages. Some results and a question added.

Let $\pi(n)$ denote the prime-counting function and let

%

$$f(n)=\left|\left\lfloor\log n-\lfloor\log n\rfloor-0.1\right\rfloor\right|\left\lfloor\frac{\left\lfloor n/\lfloor\log n-1\rfloor\right\rfloor\lfloor\log n-1\rfloor}{n}\right\rfloor\text{.}$$

%

In this paper we prove that if $n$ is an integer $\ge 60184$ and $f(n)=0$, then $\pi(n)$ does not divide $n$. We also show that if $n\ge 60184$ and $\pi(n)$ divides $n$, then $f(n)=1$. In addition, we prove that if $n\ge 60184$ and $n/\pi(n)$ is an integer, then $n$ is a multiple of $\lfloor\log n-1\rfloor$ located in the interval $[e^{\lfloor\log n-1\rfloor+1},e^{\lfloor\log n-1\rfloor+1.1}]$. This allows us to show that if $c$ is any fixed integer $\ge 12$, then in the interval $[e^c,e^{c+0.1}]$ there is always an integer $n$ such that $\pi(n)$ divides $n$.

Let $S$ denote the sequence of integers generated by the function $d(n)=n/\pi(n)$ (where $n\in\mathbb{Z}$ and $n>1$) and let $S_k$ denote the $k$th term of sequence $S$. Here we ask the question whether there are infinitely many positive integers $k$ such that $S_k=S_{k+1}$.

**Category:** Number Theory

[11] **viXra:1409.0048 [pdf]**
*replaced on 2018-06-16 09:35:52*

**Authors:** Simon Plouffe

**Comments:** 4097 Pages. This is an update of the paper of 2014.

These are all the formulas found with the help of GFUN program and some others

**Category:** Number Theory

[10] **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

[9] **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

[8] **viXra:1409.0039 [pdf]**
*replaced on 2015-02-18 12:27:35*

**Authors:** Marius Coman

**Comments:** 145 Pages. Published by Education Publishing, USA. Copyright 2015 by Marius Coman.

This book brings together sixty-two articles regarding primes submitted by the author to the preprint scientific database Vixra, papers on squares of primes, semiprimes, twin primes, sequences of primes, ways to write primes, special classes of composites, formulas for generating large primes, formulas for generating different duplets and triplets of primes, generalizations of the twin primes and de Polignac’s conjectures, generalizations of Cunningham chains, Smarandache generalized Fermat numbers, and on many other issues very much related with the study of primes. Finally, in the last eight from these collected papers, I defined a new function, the MC function, and showed some of its possible applications: for instance, I conjectured that for any pair of twin primes (p, p + 2) there exist a positive integer n of the form 15 + 18*k such that the value of Smarandache function for n is equal to p and the value of MC function for n is equal to p+2 and I also made a Diophantine analysis of few Smarandache types sequences using the MC function.

**Category:** Number Theory

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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