# Number Theory

## 1003 Submissions

 viXra:1003.0274 [pdf] submitted on 31 Mar 2010

### The New Prime Theorem (2)

Authors: Chun-Xuan Jiang
Comments: 1 pages

Using Jiang function we prove that there exist infinitely many primes P such that P1 and P2 are all prime.
Category: Number Theory

 viXra:1003.0273 [pdf] submitted on 31 Mar 2010

### The New Prime Theorem (1)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that there exist infinitely many primes P1 such that a P1 + b is prime.
Category: Number Theory

 viXra:1003.0271 [pdf] submitted on 8 Mar 2010

### Three Conjectures and Two Open Generalized Problems in Number Theory

Authors: Florentin Smarandache
Comments: 3 pages

On a Problem with Primes.
Category: Number Theory

 viXra:1003.0264 [pdf] submitted on 30 Mar 2010

### New Prime K-Tuple Theorem (6)

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 + 4n is prime.
Category: Number Theory

 viXra:1003.0263 [pdf] submitted on 30 Mar 2010

### New Prime K-Tuple Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 1 pages

Using Jiang function we prove that for every positive integer k there exist infinitely many primes P1 and P2 such that each of 1 2 jP1 + (j + 1)P2 is prime.
Category: Number Theory

 viXra:1003.0262 [pdf] submitted on 30 Mar 2010

### New Prime K-Tuple Theorem (4)

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

 viXra:1003.0261 [pdf] submitted on 30 Mar 2010

### New Prime K-Tuple Theorem (3)

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

 viXra:1003.0260 [pdf] submitted on 30 Mar 2010

### New Prime K-Tuple Theorem (2)

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

 viXra:1003.0258 [pdf] submitted on 28 Mar 2010

### New Prime K-Tuple Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jinag funciton we prove that there exist infinitely many primes P1 and P2 such that each of P1 + jP2 + j is prime and there exist infinitely many primes P1 and P2 such that each of P1 + jP2 + j is prime.
Category: Number Theory

 viXra:1003.0235 [pdf] replaced on 26 May 2010

### Justification of the Zeta Regularization Procedure for the Integrals ∫xm-Sdx

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 ∫xm-sdx via the zeta regularization of the divergent series Σxm-sdx and the zeta function ζ(m - s)
Category: Number Theory

 viXra:1003.0234 [pdf] replaced on 26 Mar 2010

### The Hardy-Littlewood Prime K-Tuple Conjecture is False

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

 viXra:1003.0233 [pdf] submitted on 7 Mar 2010

### Sequences of Numbers Involved in Unsolved Problems

Authors: Florentin Smarandache
Comments: 141 pages

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein.
Category: Number Theory

 viXra:1003.0230 [pdf] submitted on 7 Mar 2010

### Applications of Smarandache Function, and Prime and Coprime Functions

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.  In this article we are going to see that the value this function takes when n is a perfect number of the form n = 2k - 1.(2k - 1) , p = 2k - 1 being a prime number.
Category: Number Theory

 viXra:1003.0228 [pdf] submitted on 7 Mar 2010

### Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences

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

 viXra:1003.0225 [pdf] submitted on 7 Mar 2010

### Geometric Theorems, Diophantine Equations, and Arithmetic Functions

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

 viXra:1003.0220 [pdf] submitted on 7 Mar 2010

### Pluckings from the Tree of Smarandache Sequences and Functions

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

 viXra:1003.0219 [pdf] submitted on 7 Mar 2010

### Smarandache Sequences, Stereograms and Series

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

 viXra:1003.0217 [pdf] submitted on 7 Mar 2010

### Mainly Natural Numbers

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

 viXra:1003.0216 [pdf] submitted on 7 Mar 2010

### The Smarandache Function

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

 viXra:1003.0211 [pdf] submitted on 18 Mar 2010

### Goldbach Conjecture (A): Upper Bound Estimation and Lower Bound Estimation

Authors: Tong Xin Ping
Comments: 4 Pages, In Chinese

We have sieve method formula of π(N) and sieve method formula of r2(N). By these sieve method formula, we can obtain (see paper for equation)
Category: Number Theory

 viXra:1003.0199 [pdf] submitted on 6 Mar 2010

### New Classes of Codes for Cryptologists and Computer Scientists

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

 viXra:1003.0198 [pdf] submitted on 6 Mar 2010

### Six Conjectures Which Generalize or Are Related to Andrica's Conjecture

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

 viXra:1003.0189 [pdf] submitted on 16 Mar 2010

### Euclid-Euler-Jiang Prime Theorem

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

 viXra:1003.0188 [pdf] submitted on 16 Mar 2010

### There Are Infinitely Many Prime Triplets

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

 viXra:1003.0186 [pdf] submitted on 6 Mar 2010

### Inequalities for Integer and Fractional Parts

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

 viXra:1003.0180 [pdf] submitted on 6 Mar 2010

### Smarandache Type Function Obtained by Duality

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

 viXra:1003.0179 [pdf] replaced on 16 Jul 2010

### Prime Sieve Using Multiplication Operation Table

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.  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.  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. 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

 viXra:1003.0178 [pdf] submitted on 6 Mar 2010

### On Solving General Linear Equations in the Set of Natural Numbers

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 ), 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

 viXra:1003.0177 [pdf] submitted on 6 Mar 2010

### Criteria of Primality

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

 viXra:1003.0170 [pdf] submitted on 14 Mar 2010

### Diophantine Equation (See Paper) Has Infinitely Many Prime Solutions

Authors: Chun-Xuan Jiang
Comments: 18 pages

By using the arithmetic function J2n+1(ω) we prove that Diophantine equation (see paper) has infinitely many prime solutions.It is the Book proof. The J2n+1(ω) ushers in a new era in the prime numbers theory.
Category: Number Theory

 viXra:1003.0163 [pdf] submitted on 6 Mar 2010

### A Generalization of Euler's Theorem on Congruencies

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 aφ(m) = 1 (mod m)", for the case when a and m are not relatively primes.
Category: Number Theory

 viXra:1003.0153 [pdf] submitted on 6 Mar 2010

### Conjectures on Partitions of Integers as Summations of Primes

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

 viXra:1003.0151 [pdf] submitted on 6 Mar 2010

### A Method of Solving a Diophantine Equation of Second Degree with N Variables

Authors: Florentin Smarandache
Comments: 11 pages

A METHOD OF SOLVING A DIOPHANTINE EQUATION OF SECOND DEGREE WITH N VARIABLES
Category: Number Theory

 viXra:1003.0139 [pdf] submitted on 12 Mar 2010

### Gaps Among Products of m Primes

Authors: Chun-Xuan Jiang
Comments: 9 pages

Using Jiang function J2(ω) we prove gaps among products of m prime: d(x) = d(x + 1) = d(x + 5 - 3) = d(x + 7 - 3) = ... = d(x + Pn - 3) = m > 1 infinitely-often, where Pn denotes the n - th prime.
Category: Number Theory

 viXra:1003.0122 [pdf] submitted on 6 Mar 2010

### A Generalized Numeration Base

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

 viXra:1003.0121 [pdf] submitted on 6 Mar 2010

### G Add-On, Digital, Sieve, General Periodical, and Non-Arithmetic Sequences

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

 viXra:1003.0120 [pdf] submitted on 6 Mar 2010

### Another Set of Sequences, Sub-Sequences, and Sequences of Sequences

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

 viXra:1003.0118 [pdf] submitted on 6 Mar 2010

### Numerology

Authors: Florentin Smarandache
Comments: 16 pages

A collection of original sequences, open questions, and problems are mentioned below.
Category: Number Theory

 viXra:1003.0112 [pdf] submitted on 6 Mar 2010

### Convergence of a Family of Series

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

 viXra:1003.0111 [pdf] submitted on 6 Mar 2010

### A Numerical Function in the Congruence Theory

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

 viXra:1003.0107 [pdf] submitted on 6 Mar 2010

### A General Theorem for the Characterization of N Prime Numbers Simultaneously

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 , as well as I. Cucurezeanu's theorem (, p. 165), Clement's theorem, S. Patrizio's theorems , etc. Particularly, this General Theorem offers different characterizations for twin primes, for quadruple primes, etc.
Category: Number Theory

 viXra:1003.0103 [pdf] submitted on 6 Mar 2010

### About the Characteristic Function of a Set

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

 viXra:1003.0102 [pdf] submitted on 6 Mar 2010

### On Carmichaël's Conjecture

Authors: Florentin Smarandache
Comments: 4 pages

On Carmichaël's conjecture
Category: Number Theory

 viXra:1003.0095 [pdf] submitted on 6 Mar 2010

### About Bernoulli's Numbers

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 ) are well known. In this article we present a simple proof of the method from .
Category: Number Theory

 viXra:1003.0093 [pdf] submitted on 6 Mar 2010

### Bases of Solutions for Linear Congruences

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

 viXra:1003.0089 [pdf] replaced on 12 May 2010

### Complete Exposition of Non-Primes Generated from a Geometric Revolving Approach by 8x8 Sets of Related Series, and thereby ad negativo Exposition of a Systematic Pattern for the Totality of Prime Numbers

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 112 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 112. 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

 viXra:1003.0087 [pdf] submitted on 8 Mar 2010

### Santilli's Isomathematical Theory for Changing Modern Mathematics

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

 viXra:1003.0086 [pdf] submitted on 8 Mar 2010

### Fermat's Last Theorem Has Been Proved(2)

Authors: Chun-Xuan Jiang
Comments: 5 pages

In this paper we prove that it is sufficient to prove S13 + S23 = 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 S13 + S23 = 1. Fermat's last theorem has been proved.
Category: Number Theory

 viXra:1003.0084 [pdf] submitted on 8 Mar 2010

### The Approximate Solutions of Blasius Equation

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

 viXra:1003.0069 [pdf] submitted on 6 Mar 2010

### Applications of Wallis Theorem

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

 viXra:1003.0068 [pdf] submitted on 6 Mar 2010

### On Diophantine Equation X2 = 2Y4 1

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

 viXra:1003.0067 [pdf] submitted on 6 Mar 2010

### Algorithms for Solving Linear Congruences and Systems of Linear Congruences

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

 viXra:1003.0063 [pdf] submitted on 6 Mar 2010

### About Factorial Sums

Authors: Mihály Bencze, Florentin Smarandache
Comments: 3 pages

In this paper, we present some new inequalities for factorial sum.
Category: Number Theory

 viXra:1003.0061 [pdf] submitted on 6 Mar 2010

### Thirty-Six Unsolved Problems in Number Theory

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

 viXra:1003.0004 [pdf] replaced on 8 Mar 2010

### A Proof of Riemann Hypothesis Using the Growth of Mertens Function M(x)

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(x1/2+ε) for ε > 0. Also Littlewood proved that "the RH is equivalent to the statement that limx → ∞ M(x)x-1/2-ε = 0, for every ε > 0". 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