[55] **viXra:1003.0274 [pdf]**
*submitted on 31 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that there exist infinitely many primes P such that
P_{1} and P_{2} are all prime.

**Category:** Number Theory

[54] **viXra:1003.0273 [pdf]**
*submitted on 31 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove that there exist infinitely many primes P_{1} such that
a P_{1} + b is prime.

**Category:** Number Theory

[53] **viXra:1003.0271 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 3 pages

On a Problem with Primes.

**Category:** Number Theory

[52] **viXra:1003.0264 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that for every positive integer k there exist infinitely
many primes P such that each of P + 4^{n} is prime.

**Category:** Number Theory

[51] **viXra:1003.0263 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that for every positive integer k there exist infinitely
many primes P_{1} and P_{2} such that each of 1 2 jP_{1} + (j + 1)P_{2} is prime.

**Category:** Number Theory

[50] **viXra:1003.0262 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that for every positive integer k there exist infinitely
many primes P such that each of P + (2j)^{2} is prime.

**Category:** Number Theory

[49] **viXra:1003.0261 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that for every positive integer k there exist infinitely
many primes P such that each of jP + j +1 is prime.

**Category:** Number Theory

[48] **viXra:1003.0260 [pdf]**
*submitted on 30 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that for every positive integer k there exist infinitely
many primes P such that each of P + j(j + 1) is prime

**Category:** Number Theory

[47] **viXra:1003.0258 [pdf]**
*submitted on 28 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jinag funciton we prove that there exist infinitely many primes P_{1} and P_{2} such
that each of P_{1} + jP_{2} + j is prime and there exist infinitely many primes P_{1} and P_{2}
such that each of P_{1} + jP_{2} + j is prime.

**Category:** Number Theory

[46] **viXra:1003.0235 [pdf]**
*replaced on 26 May 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 15 pages

In this paper we review and try to justify some results we gave before
concerning the zeta regularization of integrals ∫x^{m-s}dx
via the zeta regularization of the divergent series
Σx^{m-s}dx and the zeta function ζ(m - s)

**Category:** Number Theory

[45] **viXra:1003.0234 [pdf]**
*replaced on 26 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 7 pages

Using Jiang function we prove Jiang prime -tuple theorem. We prove that the Hardy-Littlewood
prime-tuple conjecture is false. Jiang prime -tuple theorem can replace the Hardy-Littlewood
prime-tuple conjecture.

**Category:** Number Theory

[44] **viXra:1003.0233 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 141 pages

Over 300 sequences and many unsolved problems and conjectures
related to them are presented herein.

**Category:** Number Theory

[43] **viXra:1003.0230 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Sebastián Martín Ruiz

**Comments:** 25 pages

The Smarandache function is defined as follows:
S(n)= the smallest positive integer such that S(n)! is divisible by n. [1]
In this article we are going to see that the value this function takes when n
is a perfect number of the form n = 2^{k - 1}.(2^{k} - 1) , p = 2^{k} - 1 being a prime
number.

**Category:** Number Theory

[42] **viXra:1003.0228 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Amarnath Murthy, Charles Ashbacher

**Comments:** 219 pages

This book arose out of a collection of papers written by Amarnath Murthy. The papers
deal with mathematical ideas derived from the work of Florentin Smarandache, a man
who seems to have no end of ideas. Most of the papers were published in Smarandache
Notions Journal and there was a great deal of overlap. My intent in transforming the
papers into a coherent book was to remove the duplications, organize the material based
on topic and clean up some of the most obvious errors. However, I made no attempt to
verify every statement, so the mathematical work is almost exclusively that of Murthy.

**Category:** Number Theory

[41] **viXra:1003.0225 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** József Sándor

**Comments:** 302 pages

This book contains short notes or articles, as well as studies on several topics of
Geometry and Number theory. The material is divided into five chapters: Geometric theorems;
Diophantine equations; Arithmetic functions; Divisibility properties of numbers
and functions; and Some irrationality results. Chapter 1 deals essentially with geometric
inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have
an arithmetic character (as 9-12) on number theoretic problems in Geometry

**Category:** Number Theory

[40] **viXra:1003.0220 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Charles Ashbacher

**Comments:** 80 pages

In writing a book, one encounters and overcomes many obstacles. Not the least of which is
the occasional case of writer's block. This is especially true in mathematics where
sometimes the answer is currently and may for all time be unknown. There is nothing
worse than writing yourself into a corner where your only exit is to build a door by
solving unsolved problems. In any case, it is my hope that you will read this volume
and come away thinking that I have overcome enough of those obstacles to make the book
worthwhile. As always, your comments and criticisms are welcome. Feel free to contact
me using any of the addresses listed below, although e-mail is the preferred method.

**Category:** Number Theory

[39] **viXra:1003.0219 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Charles Ashbacher

**Comments:** 135 pages

This is the fifth book that I have written that expands on the ideas of Florentin
Smarandache. In addition, I have edited two others that also deal with the areas of
mathematics under the Smarandache Notions umbrella. All of this is a credit to the
breadth and depth of his mathematical achievement. Therefore, I once again must
commend and thank him for providing so much material to work with. I also would like
to thank J. McGray for her encouragement and patience as I struggled to make this book
a reality. The material cited in this book can be found at the website
http://www.gallup.unm.edu/~smarandache/.
The deepest thanks go to my mother Paula Ashbacher, who encouraged me to play sports,
but in the off chance that I would never learn to hit the curve ball, also insisted that I read
books. This proved to be a wise career strategy.
Finally, I would like to express my deep love for Kathy Brogla, my partner/soul
mate/best friend. So pretty and vivacious, she makes life fun, exciting and a joy to
experience every single day. She is a remarkable woman and I am so blessed to have her
in my life. Kathy is also the creator of the image on the front cover.

**Category:** Number Theory

[38] **viXra:1003.0217 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Henry Ibstedt

**Comments:** 97 pages

This book consists of a selection of papers most of which were produced
during the period 1999-2002. They have been inspired by questions raised in
recent articles in current Mathematics journals and in Florentin Smarandache's
wellknown publication Only Problems, Not Solutions.

**Category:** Number Theory

[37] **viXra:1003.0216 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** C. Dumitrescu, V. Seleacu

**Comments:** 137 pages

The function named in the title of this book is originated from the
exiled Romanian mathematician Florentin Smarandache.

**Category:** Number Theory

[36] **viXra:1003.0211 [pdf]**
*submitted on 18 Mar 2010*

**Authors:** Tong Xin Ping

**Comments:** 4 Pages, In Chinese

We have sieve method formula of π(N) and sieve method formula of r^{2}(N). By these
sieve method formula, we can obtain (see paper for equation)

**Category:** Number Theory

[35] **viXra:1003.0199 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 206 pages

Historically a code refers to a cryptosystem that deals with
linguistic units: words, phrases etc. We do not discuss such
codes in this book. Here codes are message carriers or
information storages or information transmitters which in time
of need should not be decoded or read by an enemy or an
intruder. When we use very abstract mathematics in using a
specific code, it is difficult for non-mathematicians to make use
of it. At the same time, one cannot compromise with the
capacity of the codes. So the authors in this book have
introduced several classes of codes which are explained very
non-technically so that a strong foundation in higher
mathematics is not needed. The authors also give an easy
method to detect and correct errors that occur during
transmission. Further some of the codes are so constructed to
mislead the intruder. False n-codes, whole n-codes can serve
this purpose.

**Category:** Number Theory

[34] **viXra:1003.0198 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 3 pages

TSix conjectures on pairs of consecutive primes are listed below together with
examples in each case.

**Category:** Number Theory

[33] **viXra:1003.0189 [pdf]**
*submitted on 16 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 13 pages

Santilli's prime chains: (see paper for equations) There exist infinitely many primes such that are
primes for arbitrary length .
It is the Book proof. This is a generalization of Euclid-Euler proof for the existence of
infinitely many primes. Therefore Euclid-Euler-Jiang theorem in the distribution of primes is
advanced. It is the Book theorem.

**Category:** Number Theory

[32] **viXra:1003.0188 [pdf]**
*submitted on 16 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 5 pages

Using Jiang's function we prove that there are infinitely many primes such that 3P-2 and 3P+2 are primes.

**Category:** Number Theory

[31] **viXra:1003.0186 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 8 pages

In this paper we present some new inequalities relative to
integer and functional parts.

**Category:** Number Theory

[30] **viXra:1003.0180 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** C. Dumitrescu, N. Vîrlan, Şt. Zamfir, E. Rădescu, N. Rădescu, Florentin Smarandache

**Comments:** 15 pages

In this paper we extended the Smarandache function from the set N* of
positive integers to the set Q of rational numbers.
Using the inversion formula, this function is also regarded as a generating
function. We put in evidence a procedure to construct a (numerical) function starting
from a given function in two particular cases. Also connections between the
Smarandache function and Euler's totient function as with Riemann's zeta function are
established.

**Category:** Number Theory

[29] **viXra:1003.0179 [pdf]**
*replaced on 16 Jul 2010*

**Authors:** Jongsoo Park

**Comments:** 76 pages

Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many
arithmetic progressions of length k consisting only of prime numbers. [14] Four parallel proofs of Szemer'edi's
theorem have been achieved; one by direct combinatorics, one by ergodic theory, one by hypergraph theory, and
one by Fourier analysis and additive combinatorics. Even with so many proofs, Professor T. Tao points out that
with this problem, there remains a sense that our understanding of this result is incomplete; for instance, none of
the approaches were powerful enough to detect progressions in the primes, mainly due to the sparsity of the
prime sequence. [22] Oliver Lonsdale Atkin introduced a prime sieve using irreducible binary quadratic forms
and modular arithmetic; the algorithm enumerates representations of integers by certain binary quadratic forms.
A way that uses modular arithmetic is widely known: 6n+δ, 12n+δ, 30n+δ, 60n+δ.[31] In this paper,
we assert that the composite number of the 12n+1, 5, 7, 11 series as selected by a Modular Arithmetic and
Multiplication Table are not random but consist of very structural and regular arithmetic progression groups.

**Category:** Number Theory

[28] **viXra:1003.0178 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

The utility of this article is that it establishes if the number of the natural solutions
of a general linear equation is limited or not. We will show also a method of solving,
using integer numbers, the equation ax - by = c (which represents a generalization of
lemmas 1 and 2 of [4]), an example of solving a linear equation with 3 unknowns in N,
and some considerations on solving, using natural numbers, equations with n unknowns.

**Category:** Number Theory

[27] **viXra:1003.0177 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

In this article we present four necessary and sufficient conditions for a
natural number to be prime.

**Category:** Number Theory

[26] **viXra:1003.0170 [pdf]**
*submitted on 14 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 18 pages

By using the arithmetic function J_{2n+1}(ω) we prove that Diophantine equation
(see paper) has infinitely many prime solutions.It is the Book proof. The J_{2n+1}(ω) ushers in a new
era in the prime numbers theory.

**Category:** Number Theory

[25] **viXra:1003.0163 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 7 pages

In the paragraphs which follow we will prove a result which replaces the theorem
of Euler: "If (a,m) = 1, then a^{φ(m)} = 1 (mod m)",
for the case when a and m are not relatively primes.

**Category:** Number Theory

[24] **viXra:1003.0153 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 3 pages

In this short note many conjectures on partitions of integers as summations of prime
numbers are presented, which are extension of Goldbach conjecture.

**Category:** Number Theory

[23] **viXra:1003.0151 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 11 pages

A METHOD OF SOLVING A DIOPHANTINE EQUATION
OF SECOND DEGREE WITH N VARIABLES

**Category:** Number Theory

[22] **viXra:1003.0139 [pdf]**
*submitted on 12 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 9 pages

Using Jiang function J_{2}(ω) we prove gaps among products of m prime:
d(x) = d(x + 1) = d(x + 5 - 3) = d(x + 7 - 3) = ... = d(x + P_{n} - 3) = m > 1 infinitely-often,
where P_{n} denotes the n - th prime.

**Category:** Number Theory

[21] **viXra:1003.0122 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 6 pages

A Generalized Numeration Base is defined in this
paper, and then particular cases are presented, such as Prime
Base, Square Base, m-Power Base, Factorial Base, and
operations in these bases.

**Category:** Number Theory

[20] **viXra:1003.0121 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 14 pages

Other new sequences are introduced in number theory, and for each
one a general question: how many primes each sequence has.

**Category:** Number Theory

[19] **viXra:1003.0120 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 42 pages

New sequences in number theory are showed below with
definitions, examples, solved or open questions and references for
each case.

**Category:** Number Theory

[18] **viXra:1003.0118 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 16 pages

A collection of original sequences, open questions, and problems are mentioned below.

**Category:** Number Theory

[17] **viXra:1003.0112 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

In this article we will construct a family of expressions ε(n). For each element
E(n) from ε(n), the convergence of the series Σ E(n)
can be determined in accordance to the theorems of this article.

**Category:** Number Theory

[16] **viXra:1003.0111 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 8 pages

In this paper we define a function L which
will allow us to (separately or simultaneously) generalize
many theorems from Number Theory obtained by Wilson, Fermat,
Euler, Gauss, Lagrange, Leibniz, Moser, and Sierpinski.

**Category:** Number Theory

[15] **viXra:1003.0107 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 9 pages

This article presents a necessary and
sufficient theorem for N numbers, coprime two by two, to
be prime simultaneously.
It generalizes V. Popa's theorem [3], as well as
I. Cucurezeanu's theorem ([1], p. 165), Clement's theorem,
S. Patrizio's theorems [2], etc.
Particularly, this General Theorem offers different
characterizations for twin primes, for quadruple primes,
etc.

**Category:** Number Theory

[14] **viXra:1003.0103 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 11 pages

In this paper we give a method, based on the characteristic function of a set, to
solve some difficult problems of set theory found in undergraduate studies.

**Category:** Number Theory

[13] **viXra:1003.0102 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

On Carmichaël's conjecture

**Category:** Number Theory

[12] **viXra:1003.0095 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 3 pages

Many methods to compute the sum of the first n natural numbers of the same
powers (see [4]) are well known.
In this article we present a simple proof of the method from [3].

**Category:** Number Theory

[11] **viXra:1003.0093 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 5 pages

In this article we establish some properties regarding the solutions of a linear
congruence, bases of solutions of a linear congruence, and the finding of other solutions
starting from these bases.
This article is a continuation of my article "On linear congruences".

**Category:** Number Theory

[10] **viXra:1003.0089 [pdf]**
*replaced on 12 May 2010*

**Authors:** Stein E. Johansen

**Comments:** 41 pages, Submitted to Journal of Calcutta Mathematical Society, Nov 18, 2009.

We present a certain *geometrical* interpretation of the natural numbers, where these numbers appear as *joint
products of 5- and 3-multiples* located at specified *positions* in a *revolving chamber*. Numbers without
factors 2, 3 or 5 appear at *eight* such positions, and any prime number larger than 7 manifests at one of
these eight positions after a specified amount of rotations of the chamber. Our approach determines the
sets of rotations constituting primes at the respective eight positions, as the *complements* of the sets
of rotations constituting *non-primes* at the respective eight positions. These sets of rotations constituting
non-primes are exhibited from a *basic 8x8-matrix* of the *mutual products* originating from the eight prime numbers located
at the eight positions in the *original* chamber. This 8x8-matrix is proven to generate *all* non-primes
located at the eight positions in *strict rotation regularities* of the chamber. These regularities are
expressed in relation to the multiple 11^{2} as an anchoring *reference point* and by means of convenient
*translations* between certain classes of multiples. We find the expressions of rotations generating *all*
non-primes located at *same* position in the chamber as a set of *eight related series*. The *total* set of
non-primes located at the eight positions is exposed as *eight* such sets of eight series, and with each
of the series *completely* characterized by *four simple variables* when compared to a reference series
anchored in 11^{2}. This represents a *complete* exposition of non-primes generated by a quite simple mathematical
structure. *Ad negativo* this also represents a *complete* exposition of all *prime numbers* as the union of the
eight complement sets for these eight non-prime sets of eight series.

**Category:** Number Theory

[9] **viXra:1003.0087 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 7 pages, Dedicated to the 30-th anniversary of China reform and opening

We establish the Santilli's isomathematics based on the generalization of the
modern mathematics. (more see paper)

**Category:** Number Theory

[8] **viXra:1003.0086 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 5 pages

In this paper we prove that it is sufficient to prove S_{1}^{3} + S_{2}^{3} = 1
for Fermat's last theorem
using the complex hyperbolic functions in the hypercomplex variable theory. More than 200 years
ago Euler gave a proof of S_{1}^{3} + S_{2}^{3} = 1. Fermat's last theorem has been proved.

**Category:** Number Theory

[7] **viXra:1003.0084 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 4 pages

We find Blasius function to satisfy the boundary condition f'(∞) = 1 and obtain the
approximate solutions of Blasius equation.

**Category:** Number Theory

[6] **viXra:1003.0069 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 2 pages

In this paper we present theorems and applications of Wallis theorem
related to trigonometric integrals.

**Category:** Number Theory

[5] **viXra:1003.0068 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

In this note we present a method of solving this Diophantine
equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.

**Category:** Number Theory

[4] **viXra:1003.0067 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 9 pages

In this article we determine several theorems and methods for solving linear
congruences and systems of linear congruences and we find the number of distinct
solutions. Many examples of solving congruences are given.

**Category:** Number Theory

[3] **viXra:1003.0063 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 3 pages

In this paper, we present some new inequalities for factorial sum.

**Category:** Number Theory

[2] **viXra:1003.0061 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 38 pages

Partially or totally unsolved questions in number
theory and geometry especially, such as coloration
problems, elementary geometric conjectures,
partitions, generalized periods of a number,
length of a generalized period, arithmetic and
geometric progressions are exposed.

**Category:** Number Theory

[1] **viXra:1003.0004 [pdf]**
*replaced on 8 Mar 2010*

**Authors:** Young-Mook Kang

**Comments:** 6 pages, Submitted to annals of mathematics

A study of growth of M(x) as x → ∞ is one of the most useful
approach to the Riemann hypophotesis(RH). It is very known that the RH is
equivalent to which M(x) = O(x^{1/2+ε}) for ε > 0. Also Littlewood proved that
"the RH is equivalent to the statement that
lim_{x → ∞} M(x)x^{-1/2-ε} = 0, for
every ε > 0".[1] To use growth of M(x) approaches zero as x → ∞, I simply
prove that the Riemann hypothesis is valid. Now Riemann hypothesis is not
hypothesis any longer.

**Category:** Number Theory