Number Theory

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Recent Submissions

Any replacements are listed further down

[317] viXra:1201.0048 [pdf] submitted on 2012-01-10 17:45:26

The Largest Number Ever: a New Hierarchy of Super-Hyperoperators

Authors: Marco Ripà
Comments: The paper is in Italian, 4 pages long. It is related to Graham number. Traditional Copyright "all rights reserved"

In this paper we present a super-fast hyperoperator plus a method to create a new hierarchy of hyperoperators. For comparison, applying it to a base n=2, the result will be far larger than Graham's number. Finally we show a very large number based on Graham's one.
Category: Number Theory

[316] viXra:1201.0012 [pdf] submitted on 2012-01-05 00:58:17

Prime Distribution in Pythagorean Triples(1)(the Greatest Problem in Mathematics)

Authors: Chun-Xuan Jiang
Comments: 3 Pages.

Usina Jiang function we study prime distribution in Pythagporean triples(1)
Category: Number Theory

[315] viXra:1112.0090 [pdf] submitted on 2011-12-30 16:03:25

On Equivalence Between Zeta and R-Sequence

Authors: Michael A. Bucko
Comments: 2 Pages.

Conjecture on equivalence between Zeta Riemann and R-sequence.
Category: Number Theory

[314] viXra:1112.0089 [pdf] submitted on 2011-12-30 16:05:03

On P2^n Blocker Conjecture and R2 Function

Authors: Michael A. Bucko
Comments: 2 Pages.

On p2n blocker conjecture and R2 function, ie. the conjecture based on R-sequence theory.
Category: Number Theory

[313] viXra:1112.0070 [pdf] submitted on 2011-12-23 05:58:38

The New Prime Theorems(1391)-(1440)

Authors: Chun-Xuan Jiang
Comments: 90 Pages.

Using Jiang function we prove the new prime theorems(13910-(1440)
Category: Number Theory

[312] viXra:1112.0003 [pdf] submitted on 2011-12-02 21:58:25

The New Prime Theorems 1341-1390

Authors: Chun-xuan Jiang
Comments: 90 Pages.

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMA, IAS, THES, MPIM, MSRI. Recently<Annals of Mathematics> publish the many false papers of the prime numbers to see P52-53. In this paper using Jiang function J2 (w) we prove that the new prime theorems (1341)-(1390) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution πk(N0 ,2) 1 k. This is the Book theorem
Category: Number Theory

[311] viXra:1111.0059 [pdf] submitted on 17 Nov 2011

The New Prime Theorems (1291)-(1340)

Authors: Chun-Xuan Jiang
Comments: 90 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMA, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (1291)-(1340) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[310] viXra:1111.0040 [pdf] submitted on 10 Nov 2011

The New Prime Theorems (1241)-(1290)

Authors: Chun-Xuan Jiang
Comments: 90 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMA, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (1241)-(1290) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[309] viXra:1111.0038 [pdf] submitted on 10 Nov 2011

On a Strengthened Hardy-Hilbert's Type Inequality

Authors: Guangsheng Chen
Comments: 8 pages

In this paper, by using the Euler-Maclaurin expansion for the zeta function and estimating the weight function effectively, we derive a strengthenment of a Hardy-Hilbert's type inequality proved by W.Y. Zhong. As applications, some particular results are considered. work.
Category: Number Theory

[308] viXra:1111.0027 [pdf] submitted on 4 Nov 2011

Two Proofs of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 8 Pages.

( MSC=11D04) More than one century after its formulation by the Belgian mathematician Eugene Catalan, Preda Mihailescu has solved the open problem. But, is it all ? Mihailescu's solution utilizes computation on machines, we propose here not really a proof as it is entended classically, but a resolution of an equation like the resolution of the polynomial equations of third and fourth degrees. This solution is totally algebraic and does not utilize, of course, computers or any kind of calculation.
Category: Number Theory

[307] viXra:1111.0002 [pdf] submitted on 1 Nov 2011

The New Prime Theorems (1191)-(1240)

Authors: Chun-Xuan Jiang
Comments: 90 Pages.

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMA, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (1191)-(1240) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[306] viXra:1110.0054 [pdf] submitted on 18 Oct 2011

Gauge Transformations and the Riemann Hypothesis

Authors: Thomas Evans
Comments: 9 pages

Presented is a new determination of conditions proving the Riemann Hypothesis of any global L-function, drawing heavily on conceptual and mathematical parallels from quantum theory, specifically those summarized by Bohm in his 1951 text. We present a proof of this for a special case concerning the function ζ(s) , defined by Riemann in his seminal 1859 paper, "On the number of primes less than a given number". A new method of defining a system of inverted concatenations at the simple pole(s) of a global L-function is introduced and used to finalize our proof.
Category: Number Theory

[305] viXra:1110.0045 [pdf] submitted on 14 Oct 2011

Relationship Between Irrational Constants Phi and e (Including New Equations, Possible Implications)

Authors: Prateek Goel
Comments: 6 pages.

Relationship between irrational constants Phi and e (including new equations, possible implications)
Category: Number Theory

[304] viXra:1110.0041 [pdf] submitted on 13 Oct 2011

On the Sums (See Paper) and Their Relation to the Riemann Hypothesis and the Riemannweil Formula

Authors: Jose Javier Garcia Moreta
Comments: 7 pages

We study the sums (see paper) evaluated over the zeros and the imaginary part of the zeros of the Riemann Zeta function by two methods, the first method involves the use of the Hadamard product formula for the Riemann Xi-function, the second one uses the Riemann-Weill explicit formula , which relates a sum over the imaginary part of the zeros with another sum over prime numbers , we have managed to prove that the sum (see paper)
Category: Number Theory

[303] viXra:1109.0054 [pdf] submitted on 27 Sep 2011

Expression for the Mathematical Constant e

Authors: R. G. Kulkarni
Comments: 3 pages

Mathematical constant e can be expressed in logarithmic functions. There are six expressions for e. Five of them are step functions and another one is a constant function.
Category: Number Theory

[302] viXra:1109.0049 [pdf] submitted on 25 Sep 2011

About Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

More than one century after its formulation by the Belgian mathematician Eugene Catalan, Preda Mihailescu has solved the open problem. But, is it all ? Mihailescu's solution utilizes computation on machines, we propose here not really a proof as it is entended classically, but a resolution of an equation like the resolution of the polynomial equations of third and fourth degrees. This solution is totally algebraic and does not utilize, of course, computers or any kind of calculation. (Keywords : Diophantine equations, Catalan equation ; Algebraic resolution)
Category: Number Theory

[301] viXra:1109.0029 [pdf] submitted on 9 Sep 2011

Research on Some Smarandache Problems (Vol. 7)

Authors: Huaning Liu, Jing Gao
Comments: 155 pages

This book systematically introduces the works obtained by using analytic methods on Smarandache problems. The book includes the basic knowledge of analytic number theory, mean value on some Smarandache sequences, infinite series involving some Smarandache functions, hybrid mean value of divisor function and some Smarandache functions, and so on. This book could open up the reader's perspective, and inspire the reader to these fields. We want to thank all those who have helped and encouraged us to prepare this book. Professor Wenpeng Zhang gave us the first impulse for writing this book, and have read the whole manuscript very carefully. We also thank Yanni Liu and Peng Gao for cover designs. Last but not least, we would like to thank Dr. Minh Perez for his advice and friendly collaboration.
Category: Number Theory

[300] viXra:1109.0020 [pdf] submitted on 8 Sep 2011

A Resolution of Catalan Equation

Authors: Jamel Ghanouchi
Comments: 6 pages.

[299] viXra:1109.0019 [pdf] submitted on 8 Sep 2011

About the Equation A/n = 1/x + 1/y + 1/z

Authors: Jamel Ghanouchi
Comments: 6 pages.

We consider the equation a/n = 1/x + 1/y + 1/z with (x-3n/a)(y-3n/a)(z-3n/a) and x, y, z, a and n positive integers, we establish an equivalent equation. It allows to define sequences ans series. A quick calculus leads to an impossibility, wich means that the initial equation has not solutions which is in contradiction with the fact that we know solutions, we conclude that the propositions about the solutions of this equation are undecidable for some a.
Category: Number Theory

[298] viXra:1109.0016 [pdf] submitted on 7 Sep 2011

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real ?fth Root and the Real Seventh Root of a Positive Number

Authors: Daniel Cordero Grau
Comments: 6 pages.

In this paper we give the algorithms of the real cube root, the positive fourth root, the real ?fth root and the real seventh root of a positive number. Each of the four algorithms starts with a positive number in decimal notation, then, for a non negative integer p, it writes p + 1 integers g_i and it goes through p + 1 steps in each of which it compares at most 10 pairs of integers and calculates two integers r_i and d_i, increasing (if necessary) p until r_p = 0.
Category: Number Theory

[297] viXra:1108.0040 [pdf] submitted on 24 Aug 2011

Smarandache Deconstructive Sequence

Authors: Glisic Vedran
Comments: 2 Pages

Smarandache deconstructive sequence
Category: Number Theory

[296] viXra:1108.0039 [pdf] submitted on 24 Aug 2011

Numeros Felizes e Sucessoes de Smarandache: Digressoes Com O Maple

Authors: Delfim F.M. Torres
Comments: 6 Pages, in Spanish

Dando jus a matematica experimental, mostramos como o Maple pode ser usado na investigacao matematica de algumas questoes actualmente sem resposta na Teoria dos Numeros. A tese defendida e que os alunos de um curso de Matemetica podem facilinente usar o computador como um lunar onde se excites e exercita a imaginacao.
Category: Number Theory

[295] viXra:1108.0038 [pdf] submitted on 24 Aug 2011

Smarandache Sequence of Happy Cube Numbers

Authors: Muneer Jebreel
Comments: 6 Pages

I have studied the Smarandache Happy Cube Numbers and I have got some interesting results and facts. I have discovered some open problems a bout the Happy Cube and Smarandache Happy Cube Numbers.
Category: Number Theory

[294] viXra:1107.0039 [pdf] submitted on 21 Jul 2011

Pseudo-Smarandache Functions of First and Second Kind

Authors: A. S. Muktibodh, S.T. Rathod
Comments: 9 pages

In this paper we define two kinds of Pseudo-Smarandache functions. We have investigated more than fifty terms of each pseudoSmarandache function. We have proved some interesting results and properties of these functions.
Category: Number Theory

[293] viXra:1106.0046 [pdf] submitted on 21 Jun 2011

Formula Per Trovare Numeri Primi

Authors: Raffaele Cogoni
Comments: 4 pages. In Italian

Ho scoperto la formula per la ricerca di numeri primi in data 03-11-2010, ho verificato che i primi quattro numeri sono primi, per i successivi bisogna fare la verifica.
Category: Number Theory

[292] viXra:1106.0039 [pdf] submitted on 17 Jun 2011

Fermat's Last Theorem, Goldbach's Conjecture and Riemann Hypothesis

Authors: Chun-Xuan Jiang
Comments: 6 pages. In Chinese

Fermat's last theorem, Goldbach's Conjecture and Riemann Hypothesis
Category: Number Theory

[291] viXra:1105.0002 [pdf] submitted on 1 May 2011

Riemann Hypothesis Resolution Using Nash Equilibrium to Find the Best Location of Non-Trivial Zeros and Discovering the Locality-at-a Distance (Bell's Theorem) in Mathematical Field

Authors: Pankaj Mani
Comments: 32 pages.

The established concepts here in my work here will reveal entirely a new aspect about the entire physical envelope of mathematics itself , hence raising a revolutionary question in the minds of all of us that how far mathematics is truly capable of representing/describing different physical phenomenon/scenarios. This paper will change the fundamental way mathematicians have been looking at mathematics so far in history and its extremely mysterious relationship with physics and thus clarify that why it resisted the elementary methods of mathematics in past . and the most important thing is that to comprehend this paper sense fully individual imagination of reader is extremely crucia.l It will show that how even 'points in mathematical space' are also aware of John Bell's theorem( that two separated points on the piece of paper are inter-linked a nd wellinformed about each other and hence leaving no chance for breaking symmetry of pattern and non-singularity in Nature.
Category: Number Theory

[290] viXra:1104.0057 [pdf] submitted on 19 Apr 2011

The Proofs of Binary Goldbach's Theorem Using Only Partial Primes

Authors: Chun-Xuan Jiang
Comments: 21 pages.

In 1994 we discovered the new arithmetic function J₂(ω). Using it we proved the binary Goldbach's theorem [1]. In this chapter we yield the more detailed proofs of the binary Goldbach's theorem using only partial primes.
Category: Number Theory

[289] viXra:1104.0011 [pdf] submitted on 5 Apr 2011

Generalized Fermat's Last Theorem (3)

Authors: Chun-Xuan Jiang
Comments: 4 pages.

In this paper we prove Rⁿ = y₁⁵ - y ₂⁵ has no nonzero integer solutions for n > or = 2. In 1978 using this method we had proved Fermat's last theorem [1]. But on the afternoon of July 19, 1978 this proof was disproved by Chinese mathematics institute of Academia Sinica. How tragic!
Category: Number Theory

[288] viXra:1104.0010 [pdf] submitted on 5 Apr 2011

Fermat-Catalan Equations (1)

Authors: Chun-Xuan Jiang
Comments: 4 pages.

In this paper we prove that Fermat-Catalan equations d² = a³ + c⁵ and d² = a³ + c⁷ have infinitely many coprime integer solutions.
Category: Number Theory

[287] viXra:1103.0094 [pdf] submitted on 23 Mar 2011

Brocard`s Problem. Variants of Brocard`s Problem

Authors: Martiros Khurshudyan
Comments: 2 pages.

In this article we considered an open problem. One of the problems in the list of open problems of General Number Theory, existing in [1], [2] is the Brocard`s Problem, asking to find integer values of n, for which n! + 1 = m². 'Introduction' section is dedicated to the statement of the main problem. We presented some historical overview and known facts about this problem in the 'Historical overview and known facts' section , based on information presented in the web [1], [2]. In the section 'Variants of the Problem' several variants of the Problem are presented by author based on more general n! + A = k² [4] equation and asked to find solutions for them.
Category: Number Theory

[286] viXra:1103.0092 [pdf] submitted on 23 Mar 2011

Generalized Fermat's Last Theorem

Authors: Chun-Xuan Jiang
Comments: 6 pages.

In this paper we prove (...) has no nonzero integer solutions for n ≥ 2 . We define the supercomplex number
Category: Number Theory

[285] viXra:1103.0091 [pdf] submitted on 23 Mar 2011

Fermat's Last Theorem (1)

Authors: Chun-Xuan Jiang
Comments: 6 pages.

On the afternoon of July 19, 1978 this proof was disproved by Chinese mathematics institute of Academia Sinica. How tragic! We rewrite this paper.
Category: Number Theory

[284] viXra:1103.0086 [pdf] submitted on 22 Mar 2011

A Generalisation of Fermat-Catalan Conjecture

Authors: Jamel Ghanouchi
Comments: 27 pages.

We begin with Beal equation (or Fermat-Catalan) U^c+2 = X^a+2 + Y^b+2 and establish two equivalent equations. We generalize the approach to all Fermat-Catalan equations which allows us to relate the problem to Matyasevich theorem. Our approach will lead us to propose a new conjecture concerning Fermat-Catalan equations.We propose also a proof of Beal conjecture.
Category: Number Theory

[283] viXra:1103.0081 [pdf] submitted on 21 Mar 2011

Collatz Problem and Conjecture. a Generalization of the Problem

Authors: Martiros Khurshudyan
Comments: 3 pages.

The aim of this article is presents an open problem of Mathematics. We will talk and present shortly Collatz problem and conjecture to make clear our motivation for new problem. Introduction to the original Collatz problem is given as in [1],[2],[3],[4]. From our point of view a very properly introduction to the main problem. A genaralization is proposed as well as three questions are asked to a reader at the end of article, after definition of our problem. We thought, it is possible to develop a mathematical game based on Collatz problem. We leave this idea for future works.
Category: Number Theory

[282] viXra:1103.0070 [pdf] submitted on 16 Mar 2011

Patterns Related to the Smarandache Circular Sequence Primality Problem

Authors: Marco Ripà
Comments: 19 pages

In this paper, we show the internal relations among the elements of the circular sequence (1,12,21,123,231,312,1234,3412,...). We illustrate one method to minimize the number of the "candidate prime numbers" up to a given term of the sequence. So, having chosen a particular prime divisor, it is possible to analyze only a fixed number of the smallest terms belonging to a given range, thus providing the distribution of that prime factor in a larger set of elements. Finally, we combine these results with another one, also expanding the study to a few new integer sequences related to the circular one.
Category: Number Theory

[281] viXra:1103.0038 [pdf] submitted on 12 Mar 2011

Generalized Fermat's Last Theorem Rⁿ = ...

Authors: Chun-Xuan Jiang
Comments: 4 pages

IIn this paper we prove (...) has infinitely many nonzero integer solutions. We prove (...) has no nonzero integer solutions.
Category: Number Theory

[280] viXra:1103.0016 [pdf] submitted on 5 Mar 2011

Fermat Last Theorem Controversy (6)

Authors: Chun-Xuan Jiang
Comments: 22 pages.

D.Zagier(1984) and K.Inkeri(1990) said[7] Jiang mathematics is true, but Jiang determinates the irrational numbers to be very difficult for prime exponent p>2.In 1991 Jiang studies the composite exponents n=15,21,33,...,3p and proves Fermat last theorem for prime exponent p>3[1].In 1986 Gerhard Frey places Fermat last theorem at elliptic curve ,now called a Frey curve.Andrew Wiles studies Frey curve.In 1994 Wiles proves Fermat last theorem[9,10].Conclusion:Jiang proof is direct and very simple,but Wiles proof is indirect and very complex. If China mathematicians and Academia Sinica had supported and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang had proved Fermat last theorem[1].Wiles has received many prizes and awards, he should thank China mathematicians and Academia Sinica.To support and to publish Jiang Fermat last theorem paper is prohibited in Academia Sinica. Remark. Chun-Xuan Jiang,A general proof of Fermat last theorem(Chinese),Mimeograph papers,July 1978. In this paper using circulant matrix,circulant determinant and permutation group theory Jiang had proved Fermat last theorem for odd prime exponent. (6 again)
Category: Number Theory

[279] viXra:1103.0014 [pdf] submitted on 3 Mar 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (3)

Authors: Chun-Xuan Jiang
Comments: 14 pages.

[278] viXra:1103.0010 [pdf] submitted on 3 Mar 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (6)

Authors: Chun-Xuan Jiang
Comments: 14 pages.

[277] viXra:1103.0009 [pdf] submitted on 3 Mar 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 14 pages.

[276] viXra:1103.0008 [pdf] submitted on 3 Mar 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 16 pages.

[275] viXra:1103.0004 [pdf] submitted on 2 Mar 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (2)

Authors: Chun-Xuan Jiang
Comments: 35 pages.

[274] viXra:1103.0003 [pdf] submitted on 2 Mar 2011

Fermat Last Theorem Controversy (2)

Authors: Chun-Xuan Jiang
Comments: 7 pages.

The Fermat last theorem controversy is an argument between 20th century mathematicians Jiang Chun-Xuan(1992) and Andrew Wiles(1995) over who has first proved Fermat last theorem.
Category: Number Theory

[273] viXra:1103.0001 [pdf] submitted on 1 Mar 2011

Une Généralisation de la Conjecture de Fermat-Catalan

Authors: Jamel Ghanouchi
Comments: 24 pages.

[272] viXra:1102.0058 [pdf] submitted on 28 Feb 2011

The Powers of π Are Irrational

Authors: Tim Jones
Comments: 17 pages.

Transcendence of a number implies the irrationality of powers of a number, but in the case of π there are no separate proofs that powers of π are irrational. We investigate this curiosity. Transcendence proofs for e involve what we call Hermite's technique; for π's transcendence Lindemann's adaptation of Hermite's technique is used. Hermite's technique is presented and its usage is demonstrated with irrationality proofs of e and powers of e. Applying Lindemann's adaptation to a complex polynomial, π is shown to be irrational. This use of a complex polynomial generalizes and powers of π are shown to be irrational. The complex polynomials used involve roots of i and yield regular polygons in the complex plane. One can use graphs of these polygons to visualize various mechanisms used to proof π², π³, and π⁴ are irrational. The transcendence of π and e are easy generalizations from these irrational cases.
Category: Number Theory

[271] viXra:1102.0051 [pdf] submitted on 27 Feb 2011

The Diophantine Equations A² ± M B² = Cⁿ, A³ ± M B³ = D² and Y₁⁴ ± M Y₂⁴ = R²

Authors: Chun-Xuan Jiang
Comments: 16 pages.

The Diophantine equations a² ± m b² = cⁿ , and a³ ± m b³ = d² have infinitely many nonzero integer solutions, Using the methods of infinite descent and infinite ascent we prove y₁⁴ ± m y₂⁴ = R² . Using this method you prove Beal conjecture and obtain a prize of $100,000[4]. Using this method in 1978 Jiang has proved Fermat last theorem[ Chun-Xuan Jiang,A general proof of Fermat last theorem,July 1978,Mimeograph papers].
Category: Number Theory

[270] viXra:1102.0046 [pdf] submitted on 25 Feb 2011

Jiang and Wiles Who Has First Proved Fermat Last Theorem (1)

Authors: Chun-Xuan Jiang
Comments: 16 pages.

[269] viXra:1102.0024 [pdf] submitted on 15 Feb 2011

An Analytical Approach to Polyominoes and a Solution to the Goldbach Conjecture

Authors: Aziz Sahraei
Comments: 7 pages.

Always, when viewing papers whose writers show polyominoes graphically, this question crossed my mind, are there any equations which may be given to avoid the need for drawings? Polyominoes are sometimes called by the number of faces (like triomeno or tetraomino). In this paper, I try to formulate polyomino shapes and establish a correspondence between them and polynominals. About the final part where I refer to the Goldbach conjecture, I must to say that my aim is to give a geometric representation of the proof of this conjecture so that if a special chain of subsets such as, (see paper) exists in a set Ω, where both ends of the chain include trivial subsets, and if the conjecture be true for at least one arbitrary member of this chain, then it will be true for all the other members of the chain.
Category: Number Theory

[268] viXra:1102.0017 [pdf] submitted on 11 Feb 2011

Jiang and Wiles Proofs on Fermat Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 20 pages.

1637 Fermat wrote: "It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain." part (4)
Category: Number Theory

[267] viXra:1102.0011 [pdf] submitted on 8 Feb 2011

Distribution of Prime Numbers,twin Primes and Goldbach Conjecture

Authors: Subhajit Ganguly
Comments: 5 pages.

The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.
Category: Number Theory

[266] viXra:1102.0008 [pdf] submitted on 7 Feb 2011

Jiang and Wiles Proofs on Fermat Last Theorem (3)

Authors: Chun-Xuan Jiang
Comments: 18 pages.

D.Zagier(1984) and K.Inkeri(1990) said[7]:Jiang mathematics is true,but Jiang determinates the irrational numbers to be very difficult for prime exponent p.In 1991 Jiang studies the composite exponents n=15,21,33,...,3p and proves Fermat last theorem for prime exponenet p>3[1].In 1986 Gerhard Frey places Fermat last theorem at the elliptic curve that is Frey curve.Andrew Wiles studies Frey curve. In 1994 Wiles proves Fermat last theorem[9,10]. Conclusion:Jiang proof(1991) is direct and simple,but Wiles proof(1994) is indirect and complex.If China mathematicians had supported and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang had proved Fermat last theorem.Wiles has received many prizes and awards,he should thank China mathematicians.
Category: Number Theory

[265] viXra:1101.0102 [pdf] submitted on 31 Jan 2011

Jiang and Wiles Proofs on Fermat Last Theorem(2)

Authors: Chun-Xuan Jiang
Comments: 18 pages.

D.Zagier (1984) and K.Inkeri(1990) said[7]: Jiang mathematics is true, but Jiang determinates the irrational numbers to be very difficult for prime exponent p>2. In 1991 Jiang studies the composite exponents n=15,21,33,...,3p and proves Fermat last theorem. In 1986 Gerhard Frey places Fermat last theorem at the elliptic curve that is Frey curve. Andrew Wiles studies Frey curve. In 1994 Wiles proves Fermat last theorem. Conclusion:Jiang proof is direct and simple,but Wiles proof is indirect and complex.
Category: Number Theory

[264] viXra:1101.0092 [pdf] submitted on 28 Jan 2011

On Prime Factors in Old and New Sequences of Integers

Authors: Marco Ripà
Comments: This paper is 17 pages long and the Italian version has already been published here: (http://www.rudimathematici.com/bookshelf/bookshelfdb.php).

The paper shows that the only possible prime terms of the "consecutive sequence" (1,12,123,1234,...) represent 13.33% of the total, and their structure is explicited. This outcome is then extended to every permutation of their figures. The previous result is applied to a consistent subset of elements belonging to the circular sequence (resulting from the consecutive one), identifying moreover the 31 first primes. Therefore, a criterion is illustrated (further extendible) that progressively reduces the numerousness of the "candidate prime numbers". §3.3 is devoted to the solution of a similar problem. The last section introduces a new sequence which, although much larger, has the same properties as the previous ones, and it also proposes a few open problems.
Category: Number Theory

[263] viXra:1101.0091 [pdf] submitted on 28 Jan 2011

Fermat Last Theorem and Riemann Hypothesis (6)

Authors: Chun-Xuan Jiang
Comments: 18 pages

[262] viXra:1101.0090 [pdf] submitted on 28 Jan 2011

Fermat Last Theorem and Riemann Hypothesis (5)

Authors: Chun-Xuan Jiang
Comments: 18 pages

[261] viXra:1101.0089 [pdf] submitted on 28 Jan 2011

Fermat Last Theorem and Riemann Hypothesis (4)

Authors: Chun-Xuan Jiang
Comments: 20 pages

[260] viXra:1101.0087 [pdf] submitted on 26 Jan 2011

Generation of DNA Codes & the Hexagrams of I Ching from the Principle of Existence

Authors: Huping Hu, Maoxin Wu
Comments: 6 pages

In this short paper, the authors briefly discuss their preliminary thoughts on the coding of DNA and the hexagrams of I Ching based on the principle of existence. It is shown that one may mathematically generate the DNA codes from the principle of existence. It is further shown that one may also mathematically generate the hexagrams of Chinese I Ching from the principle of existence.
Category: Number Theory

[259] viXra:1101.0086 [pdf] submitted on 26 Jan 2011

Jiang and Wiles Proofs on Fermat Last Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 21 pages

In 1637 Fermat wrote: "It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain."
Category: Number Theory

[258] viXra:1101.0071 [pdf] submitted on 22 Jan 2011

Remarks on the Function η(n)

Authors: Petru Minut
Comments: 4 pages

In 1980, F.SMARANDACHE introduced (see [5]) the function ...
Category: Number Theory

[257] viXra:1101.0070 [pdf] submitted on 22 Jan 2011

New Progress on Smarandache Problems Research

Authors: Guo Xiaoyan, Yuan Xia
Comments: 147 pages, in Chinese

New Progress on Smarandache Problems Research
Category: Number Theory

[256] viXra:1101.0051 [pdf] submitted on 16 Jan 2011

The K-Number Sieve and K-Inclusion-Exclusion Formula, Principle and Harvest

Authors: Tong Xin Ping
Comments: 3 pages

1-Number Sieve: It is the Eratosthenes'-Number Sieve and the da Silva-Sylvester formula. 2-Number Sieve: We can obtain result of Goldbach' conjecture and the number of solutions of Goldbach problem. 3-Number Sieve: We can obtain result p₃ in N= p₃+p_i P₁ and 3-Inclusion-exclusion formula.
Category: Number Theory

[255] viXra:1101.0047 [pdf] submitted on 14 Jan 2011

The New Prime Theorems (1041)-(1090)

Authors: Chun-Xuan Jiang
Comments: 117 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (1041)-(1090) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[254] viXra:1101.0032 [pdf] submitted on 7 Jan 2011

The New Prime Theorems (991)-(1040)

Authors: Chun-Xuan Jiang
Comments: 122 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (991)-(1040) contain infinitely many prime solutions and no prime solutions...
Category: Number Theory

[253] viXra:1101.0028 [pdf] submitted on 6 Jan 2011

Counter to Zhou's Criticism of Jones' Proof of the Irrationality of π and π²

Authors: Tim Jones
Comments: 5 pages

A geometric proof of the irrationality of π is given. It uses an evaluation of the area given by the product of two symmetric functions, together with bounds on the integral. The symmetric functions embed the assumption of rational π; one function is dependent on n; as the evaluation of the integral exceeds the upper bound for large n for any given rational π, a contradiction is obtained. This proof has been criticized, but here some counters to the criticism are offered.
Category: Number Theory

[252] viXra:1101.0022 [pdf] submitted on 4 Jan 2011

The New Prime Theorems (941)-(990)

Authors: Chun-Xuan Jiang
Comments: 98 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (941)-(990) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[251] viXra:1012.0047 [pdf] submitted on 23 Dec 2010

The New Prime Theorems (841)-(890)

Authors: Chun-Xuan Jiang
Comments: 95 pages.

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (841)-(890) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution (see paper). This is the Book theorem.
Category: Number Theory

[250] viXra:1012.0036 [pdf] submitted on 15 Dec 2010

A Treaty of Symmetric Function Part V Using Sum of Power for Arbitrary Arithmetic Progression for Studies of Prime Numbers that Coexist Within the Equation Formed Through a New Conjecture of Symmetric Function Rule of Division

Authors: Mohd Shukri Abd Shukor
Comments: 30 pages

A new approach in deriving Sum of Power series using reverse look up method, a method where a mathematical formulation is constructed from set of data. Faulhaber [1] derived a general equation for Power sums and calculated the terms up to (Part V)
Category: Number Theory

[249] viXra:1012.0035 [pdf] submitted on 15 Dec 2010

A Treaty of Symmetric Function Part IV Using Sums of Power for Arbitrary Arithmetic Progression to Find an Approximation for Sum of Power for Non-Integers R-th Power and Expressing Riemann's Zeta Function Into Symmetric Sums of Power Form.

Authors: Mohd Shukri Abd Shukor
Comments: 19 pages

A new approach in deriving Sum of Power series using reverse look up method, a ,method where a mathematical formulation is constructed from set of data. Faulhaber [1] derived a general equation for Power sums and calculated the terms up to (Part IV)
Category: Number Theory

[248] viXra:1012.0034 [pdf] submitted on 15 Dec 2010

A Treaty of Symmetric Function Part III an Approach in Deriving General Formulation for Alternating Sums of Power for an Arbitrary Arithmetic Progression.

Authors: Mohd Shukri Abd Shukor
Comments: 19 pages

An extension of Sum of Power formulation into alternating system. The general formulation is given as follows:
Category: Number Theory

[247] viXra:1012.0033 [pdf] submitted on 15 Dec 2010

A Treaty of Symmetric Function Part II Sums of Power for an Arbitrary Arithmetic Progression for Real Power-P

Authors: Mohd Shukri Abd Shukor
Comments: 18 pages

Sums of Power mainly deal with positive integer power p (i.e. p ε⁺ Z). In this paper, I would like to show that the sums of power that I had formulated in paper part I [1] also can be applied to the non-integer power p. The sums of power for positive non-integers (i.e. SPPNI) in this paper still adopting the same general sums of power formulation. However, the value of m has no bound and it is used as precision control. The larger the value of m used, the more accuracy the result would be.
Category: Number Theory

[246] viXra:1012.0032 [pdf] submitted on 15 Dec 2010

A Treaty of Symmetric Function Part 1 Sums of Power

Authors: Mohd Shukri Abd Shukor
Comments: 41 pages

An Approach in Deriving General Formulation for Sums of Power for an Arbitrary Arithmetic Progression and Applying the Method Formulated for Expressing Fermat's Last theorem and Riemann Zeta Function into Symmetric Function. The Generalize equation also leads to the formulation of a new set of Prime Numbers in which Mersenne and Wagstaff numbers fall under it
Category: Number Theory

[245] viXra:1012.0022 [pdf] submitted on 9 Dec 2010

Automorphic Function and Fermat's Last Theorem (6)

Authors: Chun-Xuan Jiang
Comments: 4 pages

[244] viXra:1012.0021 [pdf] submitted on 9 Dec 2010

Automorphic Function and Fermat's Last Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 5 pages

[243] viXra:1012.0010 [pdf] submitted on 2 Dec 2010

Aotomorphic Functions and Fermat's Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 7 pages

[242] viXra:1012.0009 [pdf] submitted on 2 Dec 2010

Automorphic Function and Fermat's Last Theorem (3) (Fermat's Proof of FLT)

Authors: Chun-Xuan Jiang
Comments: 5 pages

[241] viXra:1012.0008 [pdf] submitted on 2 Dec 2010

Automorphic Function and Fermat's Last Theorem(2)

Authors: Chun-Xuan Jiang
Comments: 5 pages

[240] viXra:1012.0007 [pdf] submitted on 2 Dec 2010

Automorphic Functions and Fermat's Last Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 7 pages

[239] viXra:1012.0004 [pdf] submitted on 2 Dec 2010

Goldbach' Conjecture (10): the Six Details in the Hardy-Littlewood Conjecture (A)

Authors: Tong Xin Ping
Comments: 6 pages, in Chinese

This paper is to discuss the six details in the Hardy-Littlewood Conjecture (A):...
Category: Number Theory

[238] viXra:1011.0077 [pdf] submitted on 30 Nov 2010

On "Discovering and Proving that π is Irrational"

Authors: Li Zhou
Comments: 6 pages.

We discuss the logical fallacies in an article appeared in The American Mathematical Monthly [6], and present the historical origin and motivation of the simple proofs of the irrationality of π.
Category: Number Theory

[237] viXra:1011.0047 [pdf] submitted on 21 Mar 2010

On New Functions in Number Theory

Authors: Florentin Smarandache
Comments: 98 pages, in Romanian

New functions introduced in number theory by the author are presented, studied, generalized some of them, and contributions of other mathematicians to these functions are also showed up.
Category: Number Theory

[236] viXra:1011.0045 [pdf] submitted on 21 Mar 2010

New Functions in Number Theory

Authors: Florentin Smarandache
Comments: 120 pages, in Romanian

Definitions, constructions, properties, and solved and unsolved problems on Smarandache type functions are presented in this book.
Category: Number Theory

[235] viXra:1011.0031 [pdf] submitted on 20 Mar 2010

A Generalization of an Inequality of Tchebychev

Authors: Florentin Smarandache
Comments: 1 pages

Demonstration by recurrence on m .
Category: Number Theory

[234] viXra:1011.0030 [pdf] submitted on 20 Mar 2010

A Generalization of the Inequality of Minkowski

Authors: Florentin Smarandache
Comments: 1 pages

If p is a real number...
Category: Number Theory

[233] viXra:1011.0029 [pdf] submitted on 20 Mar 2010

A Generalization of the Inequality of H&oulm;lder

Authors: Florentin Smarandache
Comments: 2 pages

One generalizes the inequality of Hödler thanks to a reasoning by recurrence. As particular cases, one obtains a generalization of the inequality of Cauchy-Buniakovski-Scwartz, and some interesting applications.
Category: Number Theory

[232] viXra:1011.0027 [pdf] submitted on 20 Mar 2010

A Generalization of the Inequality Cauchybouniakovski-Schwarz

Authors: Florentin Smarandache
Comments: 2 pages

Let us consider the real numbers...
Category: Number Theory

[231] viXra:1010.0064 [pdf] submitted on 31 Oct 2010

The New Prime Theorems (791)-(840)

Authors: Chun-Xuan Jiang
Comments: 71 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (791)-(840) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[230] viXra:1010.0049 [pdf] submitted on 20 Mar 2010

Open Questions About Concatenated Primes and Metasequences

Authors: Florentin Smarandache
Comments: 3 pages

We define a metasequence as a sequence constructed with the terms of other given sequence(s). In this short note we present some open questions on concatenated primes involved in metasequences.
Category: Number Theory

[229] viXra:1010.0048 [pdf] submitted on 20 Mar 2010

Generalization and Alternatives of Kaprekar's Routine

Authors: Florentin Smarandache
Comments: 5 pages

We extend Kaprekar's Routine for a large class of applications. We also give particular examples of this generalization as alternatives to Kaprekar's Routine and Number. Some open questions about the length of the iterations until reaching either zero or a constant or a cycle, and about the length of the cycles are asked at the end.
Category: Number Theory

[228] viXra:1010.0043 [pdf] submitted on 26 Oct 2010

Bhaskaracharya Quadratics and Spade Sequences

Authors: Nathaniel S. K. Hellerstein
Comments: 13 pages

In this article I generalize on a word problem by Bhaskaracharya. The resulting quadratics are trivial to solve; but composing them, so that they have whole number solutions, is not trivial. In this article I discover a class of sequences, which I call "Spade Sequences" as homage to a hero of detective fiction, which generate both Bhaskaracharya quadratics and their solutions. The article ends with a list of such word problems, presented as a problem set with answer key.
Category: Number Theory

[227] viXra:1010.0019 [pdf] submitted on 9 Oct 2010

Research on Number Theory and Smarandache Notions

Authors: Z. Wenpeng
Comments: 151 pages

This Book is devoted to the proceedings of the Sixth International Conference on Number Theory and Smarandache Notions held in Tianshui during April 24-25, 2010. The organizers were myself and Professor Wangsheng He from Tianshui Normal University. The conference was supported by Tianshui Normal University and there were more than 100 participants. We had one foreign guest, Professor K.Chakraborty from India. The conference was a great success and will give a strong impact on the development of number theory in general and Smarandache Notions in particular. We hope this will become a tradition in our country and will continue to grow. And indeed we are planning to organize the seventh conference in coming March which will be held in Weinan, a beautiful city of shaanxi.
Category: Number Theory

[226] viXra:1010.0017 [pdf] submitted on 8 Oct 2010

Resolution of Riemann Hypothesis

Authors: Pankaj Mani
Comments: 4 pages

Resolution of Riemann hypothesis which is true and applying the same to resolve Yang Mills mass gap theory
Category: Number Theory

[225] viXra:1010.0004 [pdf] submitted on 1 Oct 2010

The New Prime Theorem (741)-(790)

Authors: Chun-Xuan Jiang
Comments: 71 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function J₂ (ω) we prove that the new prime theorems (741)-(790) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[224] viXra:1009.0058 [pdf] submitted on 19 Sep 2010

Infinite Number

Authors: Moon Kyom
Comments: 9 pages

Added the infinite sign and the infinitesimal sign and defined an operation. The infinite calculation of number became possible. The benefits gained by infinite number is as follows.
Category: Number Theory

[223] viXra:1009.0055 [pdf] submitted on 17 Sep 2010

About Goldbach and de Polignac Conjectures

Authors: Jamel Ghanouchi
Comments: 14 pages

(MSC=11) The present algebraic development begins simply by an exposition of the data of the problem. We define the primal radius : For all x an integer greater or equal to 3, we define a primal number r for which x - r and x + r are prime numbers. We see then that Goldbach conjecture would be verified because 2x = (x + r) + (x - r).We prove that the existence of r for all x ≥ 3 can not be proven. We prove also the undecidability of the existence, for all x' an integer, of a primal radius r' for which x'+r' and r'-x' are prime numbers strictly greater than 2. de Polignac conjecture would be quickly verified because 2x' = (x' + r') - (r' - x').
Category: Number Theory

[222] viXra:1009.0054 [pdf] submitted on 17 Sep 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

[221] viXra:1009.0053 [pdf] submitted on 17 Sep 2010

An Approach of Fermat-Catalan Equations

Authors: Jamel Ghanouchi
Comments: 21 pages

We begin with Beal equation (or Fermat-Catalan) U^c+2 = X^a+2 + Y^b+2 and establish two equivalent equations. We generalize the approach to all Fermat-Catalan equations which allows us to relate the problem to Matyasevich theorem. Our approach will lead us to propose a new conjecture concerning Fermat-Catalan equations. ( MSC=11D04) Keywords : Fermat-Catalan ; Diophantine equations ; Analysis ; Series ; Fourier series ; Conjecture.
Category: Number Theory

[220] viXra:1009.0049 [pdf] submitted on 14 Sep 2010

The New Prime Theorem (691)-(740)

Authors: Chun-Xuan Jiang
Comments: 70 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (691)-(740) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[219] viXra:1009.0044 [pdf] submitted on 11 Sep 2010

The New Prime Theorem (641)-(690)

Authors: Chun-Xuan Jiang
Comments: 71 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (641)-(690) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[218] viXra:1009.0041 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Square Product Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Square Product Sequence
Category: Number Theory

[217] viXra:1009.0040 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Prime Product Sequence

Authors: Micha Fleuren
Comments: 6 pages

Factoring of the Smarandache Prime Product Sequence
Category: Number Theory

[216] viXra:1009.0039 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Mirror Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Mirror Sequence
Category: Number Theory

[215] viXra:1009.0038 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Factorial Product Sequence

Authors: Micha Fleuren
Comments: 3 pages

Factoring of the Smarandache Factorial Product Sequence
Category: Number Theory

[214] viXra:1009.0037 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Back Concatenated Cube Sequence

Authors: Micha Fleuren
Comments: 5 pages

Factoring of the Smarandache Back Concatenated Cube Sequence
Category: Number Theory

[213] viXra:1009.0036 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Back Concatenated Even Sequence

Authors: Micha Fleuren
Comments: 6 pages

Factoring of the Smarandache Back Concatenated Even Sequence
Category: Number Theory

[212] viXra:1009.0035 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Back Concatenated Odd Sequence

Authors: Micha Fleuren
Comments: 6 pages

Factoring of the Smarandache Back Concatenated Odd Sequence
Category: Number Theory

[211] viXra:1009.0034 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Back Concatenated Prime Sequence

Authors: Micha Fleuren
Comments: 5 pages

Factoring of the Smarandache Back Concatenated Prime Sequence
Category: Number Theory

[210] viXra:1009.0033 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Back Concatenated Square Sequence

Authors: Micha Fleuren
Comments: 5 pages

Factoring of the Smarandache Back Concatenated Square Sequence
Category: Number Theory

[209] viXra:1009.0032 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Concatenated Cubic Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Concatenated Cubic Sequence
Category: Number Theory

[208] viXra:1009.0031 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Concatenated Even Sequence

Authors: Micha Fleuren
Comments: 6 pages

Factoring of the Smarandache Concatenated Even Sequence
Category: Number Theory

[207] viXra:1009.0030 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Concatenated Odd Sequence

Authors: Micha Fleuren
Comments: 5 pages

Factoring of the Smarandache Concatenated Odd Sequence
Category: Number Theory

[206] viXra:1009.0029 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Concatenated Prime Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Concatenated Prime Sequence
Category: Number Theory

[205] viXra:1009.0028 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Concatenated Square Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Concatenated Square Sequence
Category: Number Theory

[204] viXra:1009.0027 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Cubic Product Sequence

Authors: Micha Fleuren
Comments: 4 pages

Factoring of the Smarandache Cubic Product Sequence
Category: Number Theory

[203] viXra:1009.0026 [pdf] submitted on 14 Mar 2010

Factoring of the Smarandache Deconstructive Sequence

Authors: Micha Fleuren
Comments: 10 pages

Factoring of the Smarandache Deconstructive Sequence
Category: Number Theory

[202] viXra:1009.0021 [pdf] submitted on 7 Sep 2010

The New Prime Theorem (591)-(640)

Authors: Chun-Xuan Jiang
Comments: 70 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (591)-(640) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[201] viXra:1009.0004 [pdf] submitted on 2 Sep 2010

New Method of Genenating of Primes and Its Application to the Goldbach's Conjecture

Authors: Kunikazu Tanaka
Comments: 21 pages

Showing how to derive new expressions of generating prime numbers to demonstrate the Goldbach's Conjecture
Category: Number Theory

[200] viXra:1008.0089 [pdf] submitted on 30 Aug 2010

The New Prime Theorem (441)-(490)

Authors: Chun-Xuan Jiang
Comments: 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (441)-(490) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[199] viXra:1008.0088 [pdf] submitted on 31 Aug 2010

Goldbach' Conjecture (9): Proved Hardy-Littlewood Conjecture (A)

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

We have inclusion-exclusion formula of π(N) and inclusion-exclusion formula of r₂(N). Make use of inclusion-exclusion formula, we can obtain Hardy-Littlewood Conjecture (A).
Category: Number Theory

[198] viXra:1008.0087 [pdf] submitted on 30 Aug 2010

The New Prime Theorem (541)-(590)

Authors: Chun-Xuan Jiang
Comments: 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (541)-(590) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[197] viXra:1008.0086 [pdf] submitted on 30 Aug 2010

The New Prime Theorem (491)-(540)

Authors: Chun-Xuan Jiang
Comments: 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. In this paper using Jiang function J₂(ω) we prove that the new prime theorems (491)-(540) contain infinitely many prime solutions and no prime solutions.From (6) we are able to find the smallest solution. π_k(N₀,2) ≥ 1. This is the Book theorem.
Category: Number Theory

[196] viXra:1008.0082 [pdf] submitted on 13 Mar 2010

A Set of Conjectures on Smarandache Sequences

Authors: Sylvester Smith
Comments: 9 pages

Searching through the Archives of the Arizona State University, I found interesting sequences of numbers and problems related to them. I display some of them, and the readers are welcome to contribute with solutions or ideas.
Category: Number Theory

[195] viXra:1008.0080 [pdf] submitted on 27 Aug 2010

The New Prime Theorem (391)-(440)

Authors: Chun-Xuan Jiang
Comments: 69 pages

Using Jiang function we prove that the new prime theorems (391)-(440) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[194] viXra:1008.0069 [pdf] submitted on 25 Aug 2010

Wandering in the World of Smarandache Numbers

Authors: A.A.K. Majumdar
Comments: 217 pages

It was in mid-nineties of the last century when I received a letter from Professor Ion Patrascu of the Fratii Buzesti College, Craiova, Romania, with lots of enclosures, introducing me with this new branch of Mathematics. Though my basic undergraduate degree is in Mathematics, my research field at that time was Operations Research and Mathematical Programming.
Category: Number Theory

[193] viXra:1008.0064 [pdf] submitted on 23 Aug 2010

Goldbach' Conjecture (8): Upper Bound Estimation of Number of Goldbach' Primes

Authors: Tong Xin Ping
Comments: 3 pages, In Chinese

This upper bound estimation prevailed over upper bound estimation of Chen Jing Run
Category: Number Theory

[192] viXra:1008.0062 [pdf] submitted on 22 Aug 2010

Smarandache Consecutive Prime Sequences (n = 1 to 100)

Authors: Robert G. Wilson V
Comments: 3 pages

"Smarandache consecutive sequences" is the nth member of the consecutive sequence, e. g. Sm(11)=1234567891011, and RSm(11)=1110987654321. Following is the prime version of "Smarandache consecutive sequences"
Category: Number Theory

[191] viXra:1008.0061 [pdf] submitted on 22 Aug 2010

Some Properties of the Pseudo-Smarandache Function

Authors: Richard Pinch
Comments: 6 pages

Charles Ashbacher [1] has posed a number of questions relating to the pseudo-Smarandache function Z(n). In this note we show that the ratio of consecutive values Z(n + 1)/Z(n) and Z(n - 1)/Z(n) are unbounded; that Z(2n)/Z(n) is unbounded; that n/Z(n) takes every integer value infinitely often; and that the series Σ_n 1/Z(n)^α is convergent for any α > 1.
Category: Number Theory

[190] viXra:1008.0054 [pdf] submitted on 20 Aug 2010

Goldbach' Conjecture (7): Five Proofs that is Under Five Assumptions Term

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

According to five assumptions, get five proofs
Category: Number Theory

[189] viXra:1008.0036 [pdf] submitted on 12 Aug 2010

On the Distribution of Prime Numbers in the Intervals Defined by the Fibonacci Numbers

Authors: J. S. Markovitch
Comments: 4 pages

The number of primes in the inclusive intervals defined by consecutive Fibonacci numbers exhibits interesting behavior between the Fibonacci numbers 55 and 196418. Specifically, starting with the interval [55, 89] through the interval [121393,196418] the ratio of the number of primes in successive intervals is a value that alternates high, low, high, low, etc.
Category: Number Theory

[188] viXra:1008.0022 [pdf] submitted on 8 Aug 2010

A Short Algebraic Proof of Fermat's Last Theorem

Authors: Morgan D. Rosenberg
Comments: 11 pages

Presented herein is a proof of Fermat's Last Theorem, which is not only short (relative to Wiles' 109 page proof), but is also performed using relatively elementary mathematics. Particularly, the binomial theorem is utilized, which was known in the time of Fermat (as opposed to the elliptic curves of Wiles' proof, which belong to modern mathematics). Using the common integer expression aⁿ + bⁿ = cⁿ for Fermat's Last Theorem, the substitutions c = b+i and b = a+j are made, where i and j are integers. Using a Taylor expansion (i.e., in the form of the binomial theorem), Fermat's Last Theorem reduces to (see paper) and what remains to be proven, from this equation, is that (see paper) only has rational solutions for n=1 and n=2. This proof is presented herein, thus proving that aⁿ + bⁿ = cⁿ only has integer solutions for a, b and c for integer values of the exponent n=1 or n=2.
Category: Number Theory

[187] viXra:1008.0021 [pdf] submitted on 8 Aug 2010

The Philosophy Error of the "Almost Prime"

Authors: Tong Xin Ping
Comments: 2 pages, In Chinese

Don't confuse quantitative change and qualitative change.
Category: Number Theory

[186] viXra:1008.0006 [pdf] submitted on 4 Aug 2010

The Philosophy Error of the Proposition "9+9"~"1+2"

Authors: Tong Xin Ping
Comments: 2 Pages. In Chinese

The method of the quantitative change can not solve the problem of the qualitative change.
Category: Number Theory

[185] viXra:1008.0001 [pdf] submitted on 1 Aug 2010

The Theory About Infinity of Simple Numbers-Twins

Authors: Valery Demidovich
Comments: 15 Pages.

The work maintenance: attempt to solve a problem about definition of set of simple numbers-twins is made. In work absolutely new approach which is based on algorithm of a sieve of Eratosfena is applied.
Category: Number Theory

[184] viXra:1007.0049 [pdf] submitted on 28 Jul 2010

Goldbach' Conjecture (6): the Chinese Remainder Theorem and Goldbach' Primes

Authors: Tong Xin Ping
Comments: 4 pages. In Chinese

By the Chinese Remainder Theorem, we can obtain Goldbach' Primes
Category: Number Theory

[183] viXra:1007.0048 [pdf] submitted on 28 Jul 2010

Goldbach' Conjecture (5): When I=1~r, the P and N Are Incongruent Modulo P_i, the P is Goldbach' Primes

Authors: Tong Xin Ping
Comments: 2 pages. In Chinese

When i=1~r, the p and N are incongruent modulo p_i, The p is Goldbach' Primes
Category: Number Theory

[182] viXra:1007.0046 [pdf] submitted on 27 Jul 2010

Goldbach' Conjecture (4): the Expression of the Number of Goldbach' Primes

Authors: Tong Xin Ping
Comments: 3 pages. In Chinese

Use the inclusion-exclusion to show that the expression of the number of Goldbach' Primes.
Category: Number Theory

[181] viXra:1007.0045 [pdf] submitted on 27 Jul 2010

Goldbach' Conjecture (3): Goldbach' Primes and Eratosthenes' Sieve Method

Authors: Tong Xin Ping
Comments: 1 pages. In Chinese

By Eratosthenes' sieve method, we can obtain Goldbach' Primes.
Category: Number Theory

[180] viXra:1007.0037 [pdf] submitted on 24 Jul 2010

Goldbach' Conjecture (2): When the P is Congruent to N Modulo P_i, the P is not Goldbach' Primes

Authors: Tong Xin Ping
Comments: 2 pages.

When the p is congruent to N modulo p_i, the p is not Goldbach' Primes.
Category: Number Theory

[179] viXra:1007.0036 [pdf] submitted on 24 Jul 2010

Goldbach' Conjecture (1): Goldbach' Primes Are Symmetric Primes

Authors: Tong Xin Ping
Comments: 2 pages.

When n/2 + x and n/2 - x or y and y + (N-y) are primes, they are Goldbach' Primes. Put it another way, The Goldbach' Primes are symmetric primes.
Category: Number Theory

[178] viXra:1007.0025 [pdf] submitted on 17 Jul 2010

The New Prime Theorem (341)-(390)

Authors: Chun-Xuan Jiang
Comments: 61 pages

Using Jiang function we prove that the new prime theorems (341)-(390) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[177] viXra:1007.0021 [pdf] submitted on 10 Jul 2010

The New Prime Theorem (191)-(240)

Authors: Chun-Xuan Jiang
Comments: 61 pages

Using Jiang function we prove that the new prime theorems (191)-(240) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[176] viXra:1007.0015 [pdf] submitted on 13 Mar 2010

A Class of Stationary Sequences

Authors: Florentin Smarandache
Comments: 3 pages

We define a class of sequences {a_n} by a₁ = a and a_n+1 = P(a_n), where P is a polynomial with real coefficients. For which a values, and for which polynomials P will these sequences be constant after a certain rank? Then we generalize it from polynomials P to real functions f. In this note, the author answers this question using as reference F. Lazebnik & Y. Pilipenko's E 3036 problem from A. M. M., Vol. 91, No. 2/1984, p. 140. An interesting property of functions admitting fixed points is obtained.
Category: Number Theory

[175] viXra:1007.0013 [pdf] submitted on 10 Jul 2010

The New Prime Theorem (291)-(340)

Authors: Chun-Xuan Jiang
Comments: 61 pages

Using Jiang function J₂(ω) we prove that the new prime theorems (291)-(340) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[174] viXra:1007.0002 [pdf] submitted on 2 Jul 2010

The New Prime Theorem (241)-(290)

Authors: Chun-Xuan Jiang
Comments: 61 pages

Using Jiang function J₂(ω) we prove that the new prime theorems (241)-(290) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[173] viXra:1006.0060 [pdf] submitted on 13 Mar 2010

A Method to Solve the Diophantine Equation a X²-B Y² + C = 0

Authors: Florentin Smarandache
Comments: 10 pages.

We consider the equation (1) ax² - by² + c = 0, with a,b ε N* and c ε Z*. It is a generalization of Pell's equation: x² -Dy² = 1. Here, we show that: if the equation has an integer solution and a.b is not a perfect square, then (1) has an infinitude of integer solutions; in this case we find a closed expression for (x_n,y_n), the general positive integer solution, by an original method. More, we generalize it for any Diophantine equation of second degree and with two unknowns.
Category: Number Theory

[172] viXra:1006.0048 [pdf] submitted on 19 Jun 2010

The New Prime Theorem (101)-(130)

Authors: Chun-Xuan Jiang
Comments: 38 pages

Using Jiang function we prove that the new prime theorems (101)-(130) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[171] viXra:1006.0047 [pdf] submitted on 19 Jun 2010

The New Prime Theorem (71)-(100)

Authors: Chun-Xuan Jiang
Comments: 38 pages

Using Jiang function we prove that the new prime theorems (71)-(100) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[170] viXra:1006.0020 [pdf] submitted on 11 Jun 2010

The New Prime Theorem (141)-(190)

Authors: Chun-Xuan Jiang
Comments: 60 pages

Using Jiang function we prove that the new prime theorems (141)-(190) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[169] viXra:1006.0016 [pdf] submitted on 11 Mar 2010

Five Properties of the Smarandache Double Factorial Function

Authors: Felice Russo
Comments: 3 pages

In this paper some properties of the Smarandache double factorial function have been analyzed.
Category: Number Theory

[168] viXra:1006.0014 [pdf] submitted on 11 Mar 2010

About Very Perfect Numbers

Authors: Mihály Bencze, Florin Popovici, Florentin Smarandache
Comments: 3 pages

In this short paper we prove that the square of an odd prime number cannot be a very perfect number.
Category: Number Theory

[167] viXra:1006.0001 [pdf] submitted on 2 Jun 2010

The New Prime Theorem (131)-(140)

Authors: Chun-Xuan Jiang
Comments: 14 pages

Using Jiang function we prove that the new prime theorems (131)-(140) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[166] viXra:1005.0109 [pdf] submitted on 11 Mar 2010

A New Equation for the Load Balance Scheduling Based on the Smarandache F-Inferior Part Function

Authors: Tatiana Tabirca, Sabin Tabirca
Comments: 6 pages

This article represents an extension of [Tabirca, 2000a]. A new equation for upper bounds is obtained based on the Smarandache f-inferior part function. An example involving upper diagonal matrices is given in order to illustrate that the new equation provide a better computation.
Category: Number Theory

[165] viXra:1005.0107 [pdf] submitted on 11 Mar 2010

On the Mean Value of the Additive Analogue of Smarandache Function

Authors: Yi Yuan, Zhang Wenpeng
Comments: 3 pages

see paper for abstract
Category: Number Theory

[164] viXra:1005.0106 [pdf] submitted on 11 Mar 2010

An Introduction to the Smarandache Square Complementary Function

Authors: Felice Russo
Comments: 13 pages

In this paper the main properties of Smarandache Square Complementary function has been analyzed. Several problems still unsolved are reported too.
Category: Number Theory

[163] viXra:1005.0105 [pdf] submitted on 11 Mar 2010

Some New Results Concerning the Smarandache Ceil Function

Authors: Sabin Tabirca, Tatiana Tabirca
Comments: 7 pages

In this article we present two new results concerning the Smarandache Ceil function. The first result proposes an equation for the number of fixed-point number of the Smarandache ceil function. Based on this result we prove that the average of the Smarandache ceil function is Θ(n) .
Category: Number Theory

[162] viXra:1005.0102 [pdf] submitted on 29 May 2010

The New Prime Theorem (45)-(70)

Authors: Chun-Xuan Jiang
Comments: 33 pages

Using Jiang function we prove that the new prime theorems (45)-(70) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[161] viXra:1005.0096 [pdf] submitted on 24 May 2010

The Sieve Method of the Number of Solutions of Goldbach Conjecture (A)

Authors: Tong Xin Ping
Comments: 3 Pages, In Chinese

We can find all solutions of Goldbach conjecture (A) ling in the closed interval [pr+1, N-pr-1], and we can obtain expression of the number of solutions of Goldbach conjecture (A).
Category: Number Theory

[160] viXra:1005.0092 [pdf] submitted on 11 Mar 2010

Seven Conjectures in Geometry and Number Theory

Authors: Florentin Smarandache
Comments: 2 pages

In this short paper we propose four conjectures in synthetic geometry that generalize Erdos-Mordell Theorem, and three conjectures in number theory that generalize Fermat Numbers.
Category: Number Theory

[159] viXra:1005.0088 [pdf] submitted on 21 May 2010

The New Prime Theorem (44)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function J₂(ω) we prove that jPⁿ + 9 - j contain infinitely many prime solutions.
Category: Number Theory

[158] viXra:1005.0087 [pdf] submitted on 21 May 2010

The New Prime Theorem (43)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that jP⁸ + k - j contain infinitely many prime solutions.
Category: Number Theory

[157] viXra:1005.0086 [pdf] submitted on 21 May 2010

The New Prime Theorem (42)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that jP⁷ + k - j contain infinitely many prime solutions.
Category: Number Theory

[156] viXra:1005.0085 [pdf] submitted on 21 May 2010

The New Prime Theorem (41)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that jP⁶ + k - j contain infinitely many prime solutions.
Category: Number Theory

[155] viXra:1005.0084 [pdf] submitted on 21 May 2010

The New Prime Theorem (40)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that jP⁵ + k - j contain infinitely many prime solutions.
Category: Number Theory

[154] viXra:1005.0083 [pdf] submitted on 21 May 2010

The New Prime Theorem (39)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that if J₂(ω) ≠ 0 then there are infinitely many primes P such that each of jP⁴ + k - j is a prime, J₂(ω) = 0 then there are finite primes P such that each of jP⁴ + k - j is a prime.
Category: Number Theory

[153] viXra:1005.0067 [pdf] submitted on 11 Mar 2010

The Smarandache P and S Persistence of a Prime

Authors: Felice Russo
Comments: 5 pages.

The Smarandache P and S persistence of a prime
Category: Number Theory

[152] viXra:1005.0064 [pdf] submitted on 15 May 2010

Santilli's Isoprime Theory

Authors: Chun-Xuan Jiang
Comments: 16 Pages

We establish the Santilli's isomathematics based on the generalization of the modern mathematics. (see paper for rest of abstract with equations)
Category: Number Theory

[151] viXra:1005.0058 [pdf] submitted on 11 Mar 2010

Partition of a Set Which Contains an Infinite Arithmetic (Respectively Geometric) Progression

Authors: Florentin Smarandache
Comments: 3 pages

We prove that for any partition of a set which contains an infinite arithmetic (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression.
Category: Number Theory

[150] viXra:1005.0054 [pdf] submitted on 11 Mar 2010

Some Smarandache Problems

Authors: Mladen V. Vassilev-Missana, Krassimir T. Atanassov
Comments: 67 pages, Book in Romanian, French and English. Proposed and solved problems for students' mathematical competitions in number theory, algebra, geometry, trigonometry, calculus.

During the five years since publishing [2], we have obtained many new results related to the Smarandache problems. We are happy to have the opportunity to present them in this book for the enjoyment of a wider audience of readers. The problems in Chapter two have also been solved and published separately by the authors, but it makes sense to collate them here so that they can be better seen in perspective as a whole, particularly in relation to the problems elucidated in Chapter one. Many of the problems, and more especially the techniques employed in their solution, have wider applicability than just the Smarandache problems, and so they should be of more general interest to other mathematicians, particularly both professional and amateur number theorists.
Category: Number Theory

[149] viXra:1005.0049 [pdf] submitted on 11 Mar 2010

Only Problems not Solutions

Authors: Florentin Smarandache
Comments: 112 pages

The development of mathematics continues in a rapid rhythm, some unsolved problems are elucidated and simultaneously new open problems to be solved appear.
Category: Number Theory

[148] viXra:1005.0047 [pdf] submitted on 11 Mar 2010

A Method of Solving Certain Nonlinear Diophantine Equations

Authors: Florentin Smarandache
Comments: 2 pages

In this paper we propose a method of solving a Nonlinear Diophantine Equation by converting it into a System of Diophantine Linear Equations.
Category: Number Theory

[147] viXra:1005.0042 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (20)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any there are infinitely many primes P such that each of jP^P₀ + j+1 is a prime.
Category: Number Theory

[146] viXra:1005.0041 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (19)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any there are infinitely many primes P such that each of P^P₀ + 4ⁿ is a prime.
Category: Number Theory

[145] viXra:1005.0040 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (18)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any there are infinitely many primes kPsuch that each of P^P₀ + (2j)² is a prime.
Category: Number Theory

[144] viXra:1005.0039 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (17)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any there are infinitely many primes kPsuch that each of P^P₀ + j(j+1) is a prime.
Category: Number Theory

[143] viXra:1005.0038 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (16)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of jP⁵ + j +1 is a prime.
Category: Number Theory

[142] viXra:1005.0037 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (15)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P⁵ + 4ⁿ is a prime.
Category: Number Theory

[141] viXra:1005.0036 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (14)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P⁵ + (2j)² is a prime.
Category: Number Theory

[140] viXra:1005.0035 [pdf] submitted on 11 May 2010

New Prime K-Tuple Theorem (13)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P⁵ + j( j +1) is a prime.
Category: Number Theory

[139] viXra:1005.0032 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (12)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of jP³ + j + 1 is a prime.
Category: Number Theory

[138] viXra:1005.0031 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (11)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P³ + 4ⁿ is a prime.
Category: Number Theory

[137] viXra:1005.0030 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (10)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P³ + (2 j)² is a prime.
Category: Number Theory

[136] viXra:1005.0029 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (9)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove that P, P¹⁵ + j(j+1)(j=1,...,7) contain no prime solutions.
Category: Number Theory

[135] viXra:1005.0028 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (8)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove that P, P⁹ + j(j+1)(j=1,...,7) contain no prime solutions.
Category: Number Theory

[134] viXra:1005.0027 [pdf] submitted on 9 May 2010

New Prime K-Tuple Theorem (7)

Authors: Chun-Xuan Jiang
Comments: 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of P³ + j( j + 1) is a prime.
Category: Number Theory

[133] viXra:1005.0025 [pdf] submitted on 10 May 2010

Proof of the 3n+1 Problem for N ≥ 1

Authors: Steffen Bode
Comments: 6 Pages.

I establish the existence of a unique binary pattern inherent to the 3n+1 step, and then use this binary pattern to prove the 3n+1 problem for all positive integers.
Category: Number Theory

[132] viXra:1005.0023 [pdf] submitted on 11 Mar 2010

Considerations on New Functions in Number Theory

Authors: Florentin Smarandache
Comments: 20 pages

In this paper a small survey is presented on eighteen new functions and four new sequences, such as: Inferior/Superior f-Part, Fractional f-Part, Complementary function with respect with another function, S-Multiplicative, Primitive Function, Double Factorial Function, S-Prime and S-Coprime Functions, Smallest Power Function.
Category: Number Theory

[131] viXra:1005.0017 [pdf] submitted on 5 May 2010

About an Identity and Its Applications

Authors: Mihály Bencze, Florentin Smarandache
Comments: 2 pages

About an Identity and its Applications
Category: Number Theory

[130] viXra:1005.0008 [pdf] submitted on 2 May 2010

The Improved of the Chen Jing Run Theorem

Authors: Tong Xin Ping
Comments: 3 Pages, In Chinese

Chen Jing Run proved that "On the representation of a large even integer as the sum of a prime and the product of at most two primes" and lower bound estimations of the number of solutions. Jiang Chun Xuan, Tong Xin Ping proved that "An even integer as the sum of a prime and the product of two primes" and compute formula of the number of solutions. This paper compares the accuracy of the three formulas
Category: Number Theory

[129] viXra:1004.0140 [pdf] submitted on 10 Mar 2010

On a Concatenation Problem

Authors: Henry Ibstedt
Comments: 13 pages.

This article has been inspired by questions asked by Charles Ashbacher in the Journal of Recreational Mathematics, vol. 29.2. It concerns the Smarandache Deconstructive Sequence. This sequence is a special case of a more general concatenation and sequencing procedure which is the subject of this study. Answers are given to the above questions. The properties of this kind of sequences are studied with particular emphasis on the divisibility of their terms by primes.
Category: Number Theory

[128] viXra:1004.0135 [pdf] submitted on 30 Apr 2010

The New Prime Theorem (38)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that there are infinitely many primes P such that each of jP³ + k - j is a prime.
Category: Number Theory

[127] viXra:1004.0134 [pdf] submitted on 30 Apr 2010

The New Prime Theorem (37)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that there are infinitely many primes P such that each of jP³ + 7 - j is a prime.
Category: Number Theory

[126] viXra:1004.0133 [pdf] submitted on 30 Apr 2010

The New Prime Theorem (36)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that there are infinitely many primes P such that each of jP³ + 5 - j is a prime.
Category: Number Theory

[125] viXra:1004.0132 [pdf] submitted on 30 Apr 2010

The New Prime Theorem (35)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that there are infinitely many primes P such that 2P³ + 1 and P³ + 2 are all prime.
Category: Number Theory

[124] viXra:1004.0131 [pdf] submitted on 30 Apr 2010

The New Prime Theorem (34)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that if J₂ (ω) ≠ 0 then there are infinitely many primes P such that each of jP² + k - j is a prime, if J₂ (ω) = 0 then there are finitely many primes P such that each of jP² + k - j is a prime.
Category: Number Theory

[123] viXra:1004.0126 [pdf] submitted on 28 Apr 2010

A Fifth Smarandache Friendly Prime Pair

Authors: Philip Gibbs
Comments: 1 page

A Smarandache friendly prime pair is a pair of prime numbers (p,q), p < q, such that the product pq is equal to the sum of all primes from p to q inclusive. Previously four such pairs were known: (2,5), (3,13), (5,31) and (7,53). A fifth one is found by a brute force search.
Category: Number Theory

[122] viXra:1004.0125 [pdf] submitted on 10 Mar 2010

On a Problem Concerning the Smarandache Friendly Prime Pairs

Authors: Felice Russo
Comments: 3 pages.

In this paper a question posed in [1] and concerning the Smarandache friendly prime pairs is analysed.
Category: Number Theory

[121] viXra:1004.0123 [pdf] submitted on 27 Apr 2010

The New Prime Theorem (33)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove x² + y⁴ (J. Friedlander and H. Iwaniec, The polynomial x² + y⁴ Captures its primes, Ann. Math., 148(1998) 945-1040)
Category: Number Theory

[120] viXra:1004.0122 [pdf] submitted on 27 Apr 2010

The New Prime Theorem (32)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove x³ + 2y³ (D. R. Heath-Brown, prime represented by x³ + 2y³, Acta Math., 186(2001)1-84).
Category: Number Theory

[119] viXra:1004.0119 [pdf] submitted on 24 Apr 2010

The New Prime Theorem (31)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove and P₁ = P⁹ ± m and P₁ = (2P)⁹ ± n
Category: Number Theory

[118] viXra:1004.0118 [pdf] submitted on 24 Apr 2010

The New Prime Theorem (30)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove and P₁ = P^P0 ± m and P₁ = (2P)^p0 ± n
Category: Number Theory

[117] viXra:1004.0117 [pdf] submitted on 24 Apr 2010

The New Prime Theorem (29)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove and P₁ = P⁵ ± m and P₁ = (2P)⁵ ± n
Category: Number Theory

[116] viXra:1004.0116 [pdf] submitted on 24 Apr 2010

The New Prime Theorem (28)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that 1P = P ± m and 1 P = 2P ± n have infinitely many
Category: Number Theory

[115] viXra:1004.0115 [pdf] submitted on 23 Apr 2010

Corrections to the wu-Sprung Potential for the Riemann Zeros and a New Hamiltonian Whose Energies Are the Prime Numbers

Authors: Jose Javier Garcia Moreta
Comments: 7 pages

We review the Wu-Sprung potential adding a correction involving a fractional derivative of Riemann Zeta function, we study a global semiclassical analysis in order to fit a Hamiltonian H=T+V fitting to the Riemann zeros and another new Hamiltonian whose energy levels are precisely the prime numbers, through these paper we use the notation log_e (x) = ln(x) = log(x) for the logarithm , also unles we specify Σ_γ h(γ) means that we sum over ALL the imaginary parts of the nontrivial zero on both the upper and lower complex plane.
Category: Number Theory

[114] viXra:1004.0111 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (27)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture P: m² +1 and m² + 3 [4].
Category: Number Theory

[113] viXra:1004.0110 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (26)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture N: x³ + y³ + z³ [4].
Category: Number Theory

[112] viXra:1004.0109 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (25)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove Hardy-Littlewood conjecture M: x³ + y³ + k [4].
Category: Number Theory

[111] viXra:1004.0108 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (24)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove Hardy-Littlewood conjecture K: x³ + k [4].
Category: Number Theory

[110] viXra:1004.0107 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (23)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture F: am² + bm+ c [4].
Category: Number Theory

[109] viXra:1004.0106 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (22)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture B: P, P + k [4].
Category: Number Theory

[108] viXra:1004.0105 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (21)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove binary Goldbach conjecture and N = P1 + ... + Pn [4]
Category: Number Theory

[107] viXra:1004.0104 [pdf] submitted on 20 Apr 2010

The New Prime Theorem (20)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that Jiang prime k -tuple theorem is true[1-3] and Hardy-Littlewood prime k -tuple conjecture is false[4-8]. The tool of additive prime number theory is basically the Hardy-Littlewood prime tuple conjecutre, but can not prove and count any prime problems[6].
Category: Number Theory

[106] viXra:1004.0088 [pdf] submitted on 18 Apr 2010

The Implicit Function in the Hardy-Littewood Conjecture

Authors: Tong Xin Ping
Comments: 6 Pages, In Chinese

The implicit function in the Hardy-Littewood conjecture
Category: Number Theory

[105] viXra:1004.0087 [pdf] submitted on 10 Mar 2010

Existence and Number of Solutions of Diophantine Quadratic Equations with Two Unknowns in Z and N

Authors: Florentin Smarandache
Comments: 2 pages.

In this short note we study the existence and number of solutions in the set of integers (Z) and in the set of natural numbers (N) of Diophantine equations of second degree with two unknowns of the general form ax² - by² = c .
Category: Number Theory

[104] viXra:1004.0071 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (19)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that such that (see paper) has infinitely many prime solutions.
Category: Number Theory

[103] viXra:1004.0070 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (18)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture E : x² + 1
Category: Number Theory

[102] viXra:1004.0069 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (17)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that such that P_n = 2 P₁P₂ ... P_n-1 has infinitely many prime solutions.
Category: Number Theory

[101] viXra:1004.0068 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (16)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that there exist infinitely many primes P such that each of (j)ⁿ P + (k - j)ⁿ is a prime.
Category: Number Theory

[100] viXra:1004.0067 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (15)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that there exist infinitely many primes P such that each of (j)³ P + (k - j)³ is a prime.
Category: Number Theory

[99] viXra:1004.0066 [pdf] submitted on 10 Apr 2010

The New Prime Theorem (14)

Authors: Chun-Xuan Jiang
Comments: 3 pages

Using Jiang function we prove that there exist infinitely many primes P such that each of (j)² P + (k - j)² is a prime.
Category: Number Theory

[98] viXra:1004.0060 [pdf] submitted on 8 Apr 2010

The New Prime Theorem (13)

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove that n x aⁿ ± 1 has infinitely many prime solutions and n x 2ⁿ ± 1 have finite prime solutions.
Category: Number Theory

[97] viXra:1004.0059 [pdf] submitted on 8 Apr 2010

The New Prime Theorem (12)

Authors: Chun-Xuan Jiang
Comments: 1 pages

Using Jiang function we prove that 3 x a³ ± 1 has infinitely many prime solutions
Category: Number Theory

[96] viXra:1004.0058 [pdf] submitted on 8 Apr 2010

The New Prime Theorem (11)

Authors: Chun-Xuan Jiang
Comments: 1 pages

Using Jiang function we prove that 2 x a² ± 1 has infinitely many prime solutions
Category: Number Theory

[95] viXra:1004.0045 [pdf] submitted on 6 Apr 2010

The New Prime Theorem (10) there Are Finite Mersenne Primes and there Are Finite Repunits Primes

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove the finite Mersenne primes and the finite repunits primes.
Category: Number Theory

[94] viXra:1004.0044 [pdf] submitted on 6 Apr 2010

The New Prime Theorem (9) there Are Finite Fermat Primes

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove the finite fermat primes.
Category: Number Theory

[93] viXra:1004.0043 [pdf] submitted on 6 Apr 2010

The Prime Principle in Clusters and Nanostructures

Authors: Chun-Xuan Jiang
Comments: 7 pages

Why we have five fingers. We suggest two principles: (1) the prime principle and (2) the symmetric principle. We prove that 1, 3, 5, 7, 11, 23, 47, and 2, 4, 6, 10, 14, 22, 46, 94 are the most stable numbers, which are the basic building-blocks in clusters and nanostructures. The prime principle is the mathematical foundations for clusters and nanosciences. It is a theory of everything.
Category: Number Theory

[92] viXra:1004.0042 [pdf] submitted on 6 Apr 2010

Prime Theorem: P₂ = AP₁ + b, Polignac Theorem and Goldbach Theorem

Authors: Chun-Xuan Jiang
Comments: 2 pages

Using Jiang function we prove prime theorem: P₂ = aP₁ + b, Polignac theorem and Goldbach theorem.
Category: Number Theory

[91] viXra:1004.0041 [pdf] submitted on 8 Mar 2010

K-Factorial

Authors: Florentin Smarandache
Comments: 1 pages

As a generalization of the factorial and double factorial one defines the kfactorial of n as the below product of all possible strictly positive factors (see paper)
Category: Number Theory

[90] viXra:1004.0040 [pdf] submitted on 8 Mar 2010

Back and Forth Factorials

Authors: Florentin Smarandache
Comments: 2 pages

Back and Forth Factorials
Category: Number Theory

[89] viXra:1004.0038 [pdf] submitted on 8 Mar 2010

Souvenirs from the Empire of Numbers

Authors: Florentin Smarandache
Comments: 20 pages

Browsing through my fifth to twelfth grade years of preoccupation for creation I discovered a notebook of Number Theory. I liked to play with numbers as Tudor Arghezi (1880-1967) - our second national Romanian poet {after the genial poet Mihai Eminescu (1850-1889)} - played with words. I was so curious and amazed by the numbers' properties. Interesting theorems, equations, and inequalities! Such fascinating people who dedicated their research to numbers, just for the sake of science! I collected many results and tried to write a handbook of mathematicians and their results.
Category: Number Theory

[88] viXra:1004.0034 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (8)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that x⁶ + 1091 has no prime solutions.
Category: Number Theory

[87] viXra:1004.0033 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (7)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 15 - j is a prime.
Category: Number Theory

[86] viXra:1004.0032 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (6)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 9 - j is a prime.
Category: Number Theory

[85] viXra:1004.0031 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + k - j is a prime.
Category: Number Theory

[84] viXra:1004.0030 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 7 - j is a prime.
Category: Number Theory

[83] viXra:1004.0029 [pdf] submitted on 4 Apr 2010

The New Prime Theorem (3)

Authors: Chun-Xuan Jiang
Comments: 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 5 - j is a prime.
Category: Number Theory

[82] viXra:1004.0028 [pdf] submitted on 5 Apr 2010

Disproofs of Riemann's Hypothesis

Authors: Chun-Xuan Jiang
Comments: 13 pages

As it is well known, the Riemann hypothesis on the zeros of the ζ(s) function has been assumed to be true in various basic developments of the 20-th century mathematics, although it has never been proved to be correct. The need for a resolution of this open historical problem has been voiced by several distinguished mathematicians. By using preceding works, in this paper we present comprehensive disproofs of the Riemann hypothesis. Moreover, in 1994 the author discovered the arithmetic function J_n(ω) that can replace Riemann's ζ(s) function in view of its proved features: if J_n(ω) ≠ 0, then the function has infinitely many prime solutions; and if J_n(ω) = 0, then the function has finitely many prime solutions. By using the Jiang J₂(ω) function we prove the twin prime theorem, Goldbach's theorem and the prime theorem of the form x² + 1. Due to the importance of resolving the historical open nature of the Riemann hypothesis, comments by interested colleagues are here solicited.
Category: Number Theory

[81] viXra:1004.0027 [pdf] submitted on 4 Apr 2010

Foundations of Santilli's Isonumber Theory

Authors: Chun-Xuan Jiang
Comments: 413 pages

In my works (see the bibliography at the end of the Preface) I often expressed the view that the protracted lack of resolution of fundamental problems in science signals the needs of basically new mathematics. This is the case, for example, for: quantitative representations of biological structures; resolution of the vexing problem of grand-unification; invariant treatment of irreversibility at the classical and operator levels; identification of hadronic constituents definable in our spacetime; achievement of a classical representation of antimatter; and other basic open problems.
Category: Number Theory

[80] viXra:1004.0020 [pdf] submitted on 8 Mar 2010

Back and Forth Summands

Authors: Florentin Smarandache
Comments: 2 pages

Back and Forth Summands
Category: Number Theory

[79] 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

[78] 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

[77] 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

[76] 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

[75] 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

[74] 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

[73] 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

[72] 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

[71] 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

[70] viXra:1003.0235 [pdf] submitted on 23 Mar 2010

Justification of the Zeta Regularization Procedure for the Integrals ∫x^m-Sdx

Authors: Jose Javier Garcia Moreta
Comments: 10 pages

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

[69] viXra:1003.0234 [pdf] submitted on 23 Mar 2010

The Hardy-Littlewood Prime K-Tuple Conjecture is False

Authors: Chun-Xuan Jiang
Comments: 6 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

[68] 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

[67] 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. [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

[66] 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

[65] 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

[64] 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

[63] 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

[62] 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

[61] 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

[60] 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

[59] 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

[58] 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

[57] 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

[56] 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

[55] 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

[54] 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

[53] 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 [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

[52] 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

[51] 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 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

[50] 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 a^φ(m) = 1 (mod m)", for the case when a and m are not relatively primes.
Category: Number Theory

[49] 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

[48] 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

[47] 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 + P_n - 3) = m > 1 infinitely-often, where P_n denotes the n - th prime.
Category: Number Theory

[46] 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

[45] 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

[44] 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

[43] 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

[42] 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

[41] 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

[40] 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 [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

[39] 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

[38] 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

[37] 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 [4]) are well known. In this article we present a simple proof of the method from [3].
Category: Number Theory

[36] 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

[35] viXra:1003.0089 [pdf] submitted on 8 Mar 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: 40 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 of 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

[34] 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

[33] 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

[32] 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

[31] 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

[30] 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

[29] 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

[28] 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

[27] 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

[26] viXra:1003.0004 [pdf] submitted on 4 Mar 2010

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

Authors: Young-Mook Kang
Comments: 5 pages

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

[25] viXra:1002.0024 [pdf] submitted on 14 Feb 2010

A Derivation of π(n) Based on a Stability Analysis of the Riemann-Zeta Function

Authors: Michael Harney, Ioannis Iraklis Haranas
Comments: 1 pages, Published: Progress in Physics, vol. 2, pp.8, 2010 .

The prime-number counting function π(n), which is significant in the prime number theorem, is derived by analyzing the region of convergence of the real-part of the Riemann-Zeta function using the unilateral z-transform. In order to satisfy the stability criteria of the z-transform, it is found that the real part of the Riemann-Zeta function must converge to the prime-counting function.
Category: Number Theory

[24] viXra:1001.0047 [pdf] submitted on 29 Jan 2010

An Approach to π(x) and Other Arithmetical Functions by Variational Principles

Authors: Jose Javier Garcia Moreta
Comments: 11 Pages.

In this paper we present a method to get the prime counting function p(x) and other arithmetical functions than can be generated by a Dirichlet series, first we use the general variational method to derive the solution for a Fredholm Integral equation of first kind with symmetric Kernel K(x,y)=K(y,x), after that we find another integral equations with Kernels K(s,t)=K(t,s) for the Prime counting function and other arithmetical functions generated by Dirichlet series, then we could find a solution for ... (see paper for full abstract)
Category: Number Theory

[23] viXra:1001.0042 [pdf] submitted on 27 Jan 2010

Zeta Regularization Applied to the Problem of Riemann Hypothesis and the Calculation of Divergent Integrals

Authors: Jose Javier Garcia Moreta
Comments: 17 Pages.

In this paper we review some results of our previous papers involving Riemann Hypothesis in the sense of Operator theory (Hilbert-Polya approach) and the application of the negative values of the Zeta function ... (see paper for full abstract)
Category: Number Theory

[22] viXra:1001.0039 [pdf] submitted on 26 Jan 2010

A Comment on Mathematical Methods to Deal with Divergent Series and Integrals

Authors: Jose Javier Garcia Moreta
Comments: 14 Pages.

In this paper we study the methods of Borel and Nachbin resummation applied to the solution of integral equation with Kernels K(yx) , the resummation of divergent series and the possible application to Hadamard finite-part integral and distribution theory.
Category: Number Theory

[21] viXra:1001.0038 [pdf] submitted on 26 Jan 2010

A Note on the Mellin Convolution of Functions and Its Relation to Riesz Criterion and Riemann Hypothesis

Authors: Jose Javier Garcia Moreta
Comments: 6 Pages.

In this paper we study how the Mellin convolution of functions f and g ( f * g ) and how is related to the Riesz criterion for the Riemann Hypothesis, the idea is to stablish a Fredholm integral equation of First kind for the Riesz function and the Hardy function.
Category: Number Theory

[20] viXra:0912.0043 [pdf] submitted on 19 Dec 2009

Imanol's Numbers

Authors: Imanol Pérez
Comments: 2 Pages.

Imanol's numbers are those that the sum of their digits is 2, 3, 5, 6, 8 or 9.
Category: Number Theory

[19] viXra:0912.0040 [pdf] submitted on 18 Dec 2009

Expansión (1/x+2/x.......+a/x)ⁿ

Authors: Imanol Pérez
Comments: 2 Pages. In Spanish

Expansion of (1/x+2/x.......+a/x)ⁿ
Category: Number Theory

[18] viXra:0912.0030 [pdf] submitted on 12 Dec 2009

Diophantine Equation 1^N + 2^N + ...+ (M 1)^N +M^N = (M + 1)^N

Authors: Arkoprobho Chakraborty
Comments: 13 pages.

Erdos had conjectured that the equation of the title had no solutions in natural numbers except the trivial 1¹ + 2¹ = 3¹. Moser (1953) had shown that there are no solutions for M+1 < 10^10⁶. Butske et al (1993) had further shown that there are no solutions for M+1 < 9.3x10⁶. In this paper I show that a solution to this equation cannot exist for any value of M > 2 hence proving Erdos' conjecture. This is achieved using elementary number theoretic methods employing congruences and well-known identities.
Category: Number Theory

[17] viXra:0911.0002 [pdf] submitted on 2 Nov 2009

Infinity of Mersenne Prime Number.

Authors: Kazuya Kawai
Comments: 2 pages

The mersenne prime number exists in infinity.
Category: Number Theory

[16] viXra:0910.0012 [pdf] submitted on 9 Oct 2009

A New Formula for the Sum of the Sixth Powers of Fibonacci Numbers

Authors: Hideyuki Ohtsuka, Shigeru Nakamura
Comments: 3 Pages, Published: Congressus Numerantum, Proceedings of the Thirteenth Conference on Fibonacci Numbers and their Applications, Vol. 201, pp.297-300 (2010).

Sloane's On-Line Encyclopedia of Integer Sequences incorrectly states a lengthy formula for the sum of the sixth powers of the first n Fibonacci numbers. In this paper we prove a more succinct formulation. We also provide an analogue for the Lucas numbers. Finally, we prove a divisibility result for the sum of certain even powers of the first n Fibonacci numbers.
Category: Number Theory

[15] viXra:0909.0034 [pdf] submitted on 14 Sep 2009

On Strategies Towards the Riemann Hypothesis :Fractal Supersymmetric QM and a Trace Formula

Authors: Carlos Castro
Comments: 20 Pages. This article appeared in the Int. Jour. of Geom. Methods of Modern Physics, 4, no. 5 (2007) 881-895.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s_n = 1/2 + iλ_n. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros ) by recurring to P a weighted superposition of Weierstrass functions ΣW(x,p,D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit completion relation ( "trace formula") related to a superposition of eigenfunctions of these scaling-like operators is defined. If the completion relation is satisfied this could be another test of the Riemann Hypothesis. In an appendix we briefly describe our recent findings showing why the Riemann Hypothesis is a consequence of CT -invariant Quantum Mechanics, because < Ψ_s | CT | Ψ_s > ≠ 0 where s are the complex eigenvalues of the scaling-like operators.
Category: Number Theory

[14] viXra:0908.0098 [pdf] submitted on 26 Aug 2009

The Riemann Hypothesis is a Consequence of CT-Invariant Quantum Mechanics

Authors: Carlos Castro
Comments: 17 pages, This article appeared in the Int. Jour. of Geom. Methods of Modern Physics vol 5, no. 1, February 2008

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s_n = 1/2 + iλ_n. By constructing a continuous family of scaling-like operators involving the Gauss-Jacobi theta series and by invoking a novel CT-invariant Quantum Mechanics, involving a judicious charge conjugation C and time reversal T operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.
Category: Number Theory

[13] viXra:0908.0091 [pdf] submitted on 24 Aug 2009

Polynomials with Rational Roots that Differ by a Non-Zero Constant

Authors: Philip Gibbs
Comments: 6 pages

The problem of finding two polynomials P(x) and Q(x) of a given degree n in a single variable x that have all rational roots and differ by a non-zero constant is investigated. It is shown that the problem reduces to considering only polynomials with integer roots. The cases n < 4 are solved generically. For n = 4 the case of polynomials whose roots come in pairs (a,-a) is solved. For n = 5 an infinite number of inequivalent solutions are found with the ansatz P(x) = -Q(-x). For n = 6 an infinite number of solutions are also found. Finally for n = 8 we find solitary examples.
Category: Number Theory

[12] viXra:0908.0079 [pdf] submitted on 21 Aug 2009

On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity and Renormalization Group

Authors: Carlos Castro
Comments: 33 pages, This article will appear in the Int. J. of Geom. Methods in Mod Phys vol 7, no. 1 (2010)

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert-Polya proposal to find an operator whose spectrum reproduces the ordinates ρ_n (imaginary parts) of the zeta zeros in the critical line : s_n = 1/2 + iρ_n A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels E_n = ρ_n, when the boundary conditions Ψ_E (x = -∞) = ± Ψ_E (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation x = x(V) = π/2(dN(V)/dV), where the functional form of N(V) is given by the full-fledged Riemann-von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the O(E^-n) terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large x (O(1/logx)) has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the Renormalization Group program.
Category: Number Theory

[11] viXra:0908.0050 [pdf] submitted on 10 Aug 2009

A Proof for Goldbach's Conjecture

Authors: Hamid V. Ansari
Comments: 5 pages

For a large even number there are a large number of pairs of odd numbers sum of the members of each being the even number. We eliminate those pairs that none of the members of each of them is prime and show that the number of the remaining pairs is still large. The process of proof shows that there can be no drop to zero in the function of the number of the mentioned prime pairs.
Category: Number Theory

[10] viXra:0907.0024 [pdf] submitted on 20 Jul 2009

Adjugates of Diophantine Quadruples

Authors: Philip Gibbs
Comments: 7 pages. Published in INTEGERS 10 (2010), 201-209, (The Electronic Journal of Combinatorial Number Theory)

Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of the set placed in the diagonal and with corresponding roots off-diagonal. In the case of quadruples, Jacobi's theorem for the minors of the adjugate matrix can be used to show that up to eight new Diophantine quadruples can be formed from the adjugate matrices with various combinations of signs for the roots. We call these adjugate quadruples.
Category: Number Theory

[9] viXra:0904.0003 [pdf] submitted on 7 Apr 2009

On the Consecutive Integers N+i-1 = (I+1)P_i

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

By using the Jiang's function J₂(ω) we prove that there exist infinitely many integers n such that n = 2P₁, n+1 = 3P₂, ..., n+k-1 = (k+1)P_k are all composites for arbitrarily long k, where P₁, P₂, ..., P_k are all primes. This result has no prior occurrence in the history of number theory.
Category: Number Theory

[8] viXra:0904.0001 [pdf] submitted on 6 Apr 2009

On the Foundamental Theorem in Arithmetic Progession of Primes

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

Using Jiang function we prove the foundamental theorem in arithmetic progression of primes. The primes contain only k < P_g+1 long arithmetic progressions, but the primes have no k > P_g+1 long arithmetic progressions. Terence Tao is recipient of 2006 Fields medal. Green and Tao proved that the primes contain arbitrarily long arithmetic progressions which is absolutely false. They do not understand the arithmetic progression of primes.
Category: Number Theory

[7] viXra:0901.0003 [pdf] submitted on 14 Jan 2009

A Revision to Gödel's Incompleteness Theorem by Neutrosophy

Authors: Fu Yuhua, Fu Anjie
Comments: recovered from sciprint.org

According to Smarandache's neutrosophy, the Gödel's incompleteness theorem contains the truth, the falsehood, and the indeterminacy of a statement under consideration. It is shown in this paper that the proof of Gödel's incompleteness theorem is faulty, because all possible situations are not considered (such as the situation where from some axioms wrong results can be deducted, for example, from the axiom of choice the paradox of the doubling ball theorem can be deducted; and many kinds of indeterminate situations, for example, a proposition can be proved in 9999 cases, and only in 1 case it can be neither proved, nor disproved). With all possible situations being considered with Smarandache's neutrosophy, the Gödel's Incompleteness theorem is revised into the incompleteness axiom: Any proposition in any formal mathematical axiom system will represent, respectively, the truth (T), the falsehood (F), and the indeterminacy (I) of the statement under consideration, where T, I, F are standard or non-standard real subsets of ]-0, 1+[ . With all possible situations being considered, any possible paradox is no longer a paradox. Finally several famous paradoxes in history, as well as the so-called unified theory, ultimate theory and so on are discussed.
Category: Number Theory

[6] viXra:0901.0002 [pdf] submitted on 3 Jan 2009

The Chinese Remainder Theorem .Goldbach's Conjecture (A) .Hardy-Littewood's Conjecture (A)

Authors: Tong Xin Ping
Comments: recovered from sciprint.org

N = p_i + (N-p_i) = p+ (N-p). If p is congruent to N modulo p_i, Then (N-p) is a composite integer, When i = 1, 2,..., r, if p and N are incongruent modulo p_i, Then p and (N-p) are solutions of Goldbach's Conjecture (A); By Chinese Remainder Theorem we can calculate the primes and solutions of Goldbach's Conjecture (A) with different system of congruence; The (N-p) must have solution of Goldbach's Conjecture (A), The number of solutions of Goldbach's Conjecture (A) is increasing as N → ∞, and finding unknown particulars for Hardy-Littewood's Conjecture (A).
Category: Number Theory

[5] viXra:0812.0009 [pdf] submitted on 29 Dec 2008

Riemann Paper (1859) is False

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

In 1859 Riemann defined the zeta function ζ(s). From Gamma function he derived the zeta function with Gamma function ζ-bar(s). ζ-bar(s) and ζ(s) are the two different functions. It is false that ζ-bar(s) replaces ζ(s). Therefore Riemann hypothesis (RH) is false. The Jiang function J(ω) can replace RH.
Category: Number Theory

[4] viXra:0812.0004 [pdf] submitted on 9 Dec 2008

Jiang's Function J_n+1(ω) in Prime Distribution

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

Jiang's function J_n+1(ω) in prime distribution
Category: Number Theory

[3] viXra:0810.0002 [pdf] submitted on 1 Oct 2008

The Simplest Proofs of Both Arbitrarily Long Arithmetic Progressions of Primes

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

Using Jiang functions...
Category: Number Theory

[2] viXra:0809.0002 [pdf] submitted on 8 Sep 2008

Santilli's Isoprime Theory

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

We establish the Santilli's isomathematics based on the generalization of the modern mathematics. Isomultiplication...
Category: Number Theory

[1] viXra:0807.0005 [pdf] submitted on 13 Jul 2008

A Proof of "Goldbach's Conjecture"

Authors: Roger Ellman
Comments: recovered from sciprint.org

Every even number greater than two can be expressed as the sum of two primes.
Category: Number Theory

Recent Replacements

[78] viXra:1201.0048 [pdf] replaced on 2012-01-14 13:08:28

The Largest Number Ever

[77] viXra:1201.0012 [pdf] replaced on 2012-01-09 18:23:45

Prime Distribution in Pythagorean Triples

Authors: chun-xan jiang
Comments: 3 Pages.

using fiang function we study the prime distribution in pythagorean triples
Category: Number Theory

[76] viXra:1201.0012 [pdf] replaced on 2012-01-09 08:34:36

Prime.dictributionn in Pythagorean Triples(1)

Authors: chun-xan jiang
Comments: 3 Pages.

using jiang function we study the prime distribution triples
Category: Number Theory

[75] viXra:1201.0012 [pdf] replaced on 2012-01-08 22:09:25

Prime Distribution in Pythagorean Triples

Authors: chun-xan jiang
Comments: 3 Pages.

using jiang function we study the prime distribution in pythagorean triples
Category: Number Theory

[74] viXra:1111.0027 [pdf] replaced on 2011-12-14 16:49:31

Two Proofs of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 12 Pages. No comment.

[73] viXra:1111.0027 [pdf] replaced on 13 Nov 2011

Two Proofs of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 12 Pages.

[72] viXra:1111.0027 [pdf] replaced on 10 Nov 2011

Two Proofs of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 12 Pages.

[71] viXra:1110.0051 [pdf] replaced on 4 Nov 2011

The New Prime Theorems (1141)-(1190)

Authors: Chun-Xuan Jiang
Comments: 90 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function we prove that the new prime theorems (1141)-(1190) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution. This is the Book theorem.
Category: Number Theory

[70] viXra:1110.0032 [pdf] replaced on 3 Nov 2011

The New Prime Theorems (1091)-(1140)

Authors: Chun-Xuan Jiang
Comments: 90 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No great mathematicians study prime problems and prove Riemann hypothesis in AIM, CLAYMI, IAS, THES, MPIM, MSRI. In this paper using Jiang function we prove that the new prime theorems (1091)-(1140) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution. This is the Book theorem.
Category: Number Theory

[69] viXra:1109.0049 [pdf] replaced on 27 Sep 2011

About Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 4 pages

[68] viXra:1109.0020 [pdf] replaced on 21 Sep 2011

A Resolution of Catalan Equation

Authors: Jamel Ghanouchi
Comments: 7 pages.

[67] viXra:1109.0020 [pdf] replaced on 16 Sep 2011

A Resolution of Catalan Equation

Authors: Jamel Ghanouchi
Comments: 4 pages.

[66] viXra:1109.0019 [pdf] replaced on 14 Sep 2011

About the Equation A/n = 1/x + 1/y + 1/z

Authors: Jamel Ghanouchi
Comments: 6 pages.

[65] viXra:1109.0019 [pdf] replaced on 9 Sep 2011

About the Equation A/n = 1/x + 1/y + 1/z

Authors: Jamel Ghanouchi
Comments: 7 pages.

[64] viXra:1109.0016 [pdf] replaced on 7 Oct 2011

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real Fifth Root and the Real Seventh Root of a Positive Number

Authors: Daniel Cordero Grau
Comments: 6 pages.

In this paper we give the algorithms of the real cube root, the positive fourth root, the real fifth root and the real seventh root of a positive number. Each of the four algorithms starts with a positive number in decimal notation, then, for a non negative integer p, it writes p + 1 integers g_i and it goes through p + 1 steps in each of which it compares at most 10 pairs of integers and calculates two integers r_i and d_i, increasing (if necessary) p until r_p = 0.
Category: Number Theory

[63] viXra:1109.0016 [pdf] replaced on 1 Oct 2011

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real Fifth Root and the Real Seventh Root of a Positive Number

Authors: Daniel Cordero Grau
Comments: 6 pages.

[62] viXra:1109.0016 [pdf] replaced on 21 Sep 2011

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real Fifth Root and the Real Seventh Root of a Positive Number

Authors: Daniel Cordero Grau
Comments: 6 pages.

[61] viXra:1109.0016 [pdf] replaced on 12 Sep 2011

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real Fifth Root and the Real Seventh Root of a Positive Number

Authors: Daniel Cordero Grau
Comments: 6 pages.

[60] viXra:1104.0011 [pdf] replaced on 8 Apr 2011

Generalized Fermat's Last Theorem (3) Rⁿ = y₁⁵ − y₂⁵

Authors: Chun-Xuan Jiang
Comments: 4 pages.

In this paper we prove Rⁿ = y₁⁵ − y₂⁵ has no nonzero integer solutions for n ≥ 2. In 1978 using this method we had proved Fermat's last theorem [1]. But on the afternoon of July 19, 1978 this proof was disproved by Chinese mathematics institute of Academia Sinica. How tragic!
Category: Number Theory

[59] viXra:1103.0092 [pdf] replaced on 8 Apr 2011

Generalized Fermat's Last Theorem (2) Rⁿ = y₁⁴ − y₂⁴

Authors: Chun-Xuan Jiang
Comments: 6 pages.

In this paper we prove Rⁿ = y₁ ⁴ − y₂⁴ has no nonzero integer solutions for n ≥ 2.
Category: Number Theory

[58] viXra:1103.0086 [pdf] replaced on 2011-12-14 16:34:55

A Generalization of Fermat-Catalan Conjecture

Authors: Jamel Ghanouchi
Comments: 32 Pages. No comment.

We begin with Beal equation (or Fermat-Catalan) $U^{c+2}=X^{a+2}+Y^{b+2}$ and establish two equivalent equations. We generalize the approach to all Fermat-Catalan equations which allows us to relate the problem to Matyasevich theorem. Our approach will lead us to propose a new conjecture concerning Fermat-Catalan equations.
Category: Number Theory

[57] viXra:1103.0070 [pdf] replaced on 4 Dec 2011

Patterns Related to the Smarandache Circular Sequence Primality Problem

Authors: Marco Ripà
Comments: 21 Pages.

In this paper, we show the internal relations among the elements of the circular sequence (1,12,21,123,231,312,1234,2341,...). We illustrate one method to minimize the number of the "candidate prime numbers" up to a given term of the sequence. So, having chosen a particular prime divisor, it is possible to analyze only a fixed number of the smallest terms belonging to a given range, thus providing the distribution of that prime factor in a larger set of elements. Finally, we combine these results with another one, also expanding the study to a few new integer sequences related to the circular one.
Category: Number Theory

[56] viXra:1103.0001 [pdf] replaced on 2011-12-14 16:30:41

Une Généralisation de la Conjecture de Fermat-Catalan

Authors: Jamel Ghanouchi
Comments: 32 Pages. In French.

[55] viXra:1103.0001 [pdf] replaced on 5 Mar 2011

Une Généralisation de la Conjecture de Fermat-Catalan

Authors: Jamel Ghanouchi
Comments: 27 pages.

[54] viXra:1102.0051 [pdf] replaced on 8 Apr 2011

The Diophantine Equations A² ± M B² = Cⁿ, A³ ± M B³ = D² and Y₁⁴ ± M Y₂⁴ = R²

Authors: Chun-Xuan Jiang
Comments: 9 pages.

[53] viXra:1101.0092 [pdf] replaced on 4 Mar 2011

On Prime Factors in Old and New Sequences of Integers

Authors: Marco Ripà
Comments: This paper is 17 pages long and the Italian version has already been published here: (http://www.rudimathematici.com/bookshelf/bookshelfdb.php).

The paper shows that the only possible prime terms of the "consecutive sequence" (1,12,123,1234...) represent of the total, and their structure is explicited. This outcome is then extended to every permutation of their figures. The previous result is applied to a consistent subset of elements belonging to the circular sequence (resulting from the consecutive one), identifying moreover the 31 first primes. Therefore, a criterion is illustrated (further extendible) that progressively reduces the numerousness of the "candidate prime numbers". Section 3.3 is devoted to the solution of a similar problem. The last section introduces a new sequence which, although much larger, has the same properties as the previous ones, and it also proposes a few open problems.
Category: Number Theory

[52] viXra:1012.0032 [pdf] replaced on 19 Nov 2011

A Treaty of Symmetric Function Part 1 Sums of Power

Authors: Mohd Shukri Abd Shukor
Comments: 47 pages

Sum of Power had gathered interest of many classical mathematicians for more than two thousand years ago. The quests of finding sum of power or discrete sum of numerical power can be traced back from the time of Archimedes in third BC then to Faulhaber in the sixteen century. Until today there is no closed form sums of power formulation for an arithmetic progression has been found. Many mathematicians were involved in this research and many approaches have been introduced but none is found to be conclusive. The generalized equation for sums of power discovered in this research has been compared to Faulhaber's sums of power for integers and it is found that this new generalized equation can be used for both integers and arithmetic progression, thus offering a new frontier in studying symmetric function, Fermat's last theorem, Riemman's Zeta function etc.
Category: Number Theory

[51] viXra:1012.0022 [pdf] replaced on 9 Jan 2011

Automorphic Function and Fermat's Last Theorem (6)

Authors: Chun-Xuan Jiang
Comments: 24 pages

[50] viXra:1012.0021 [pdf] replaced on 9 Jan 2011

Automorphic Function and Fermat's Last Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 25 pages

[49] viXra:1012.0010 [pdf] replaced on 9 Jan 2011

Aotomorphic Functions and Fermat's Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 27 pages

[48] viXra:1012.0010 [pdf] replaced on 15 Dec 2010

Aotomorphic Functions and Fermat's Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 7 pages

[47] viXra:1012.0009 [pdf] replaced on 9 Jan 2011

Automorphic Function and Fermat's Last Theorem (3) (Fermat's Proof of FLT)

Authors: Chun-Xuan Jiang
Comments: 25 pages

[46] viXra:1012.0009 [pdf] replaced on 15 Dec 2010

Automorphic Function and Fermat's Last Theorem (3) (Fermat's Proof of FLT)

Authors: Chun-Xuan Jiang
Comments: 5 pages

[45] viXra:1012.0008 [pdf] replaced on 9 Jan 2011

Automorphic Function and Fermat's Last Theorem(2)

Authors: Chun-Xuan Jiang
Comments: 25 pages

[44] viXra:1012.0008 [pdf] replaced on 15 Dec 2010

Automorphic Function and Fermat's Last Theorem(2)

Authors: Chun-Xuan Jiang
Comments: 5 pages

[43] viXra:1012.0007 [pdf] replaced on 12 Jan 2011

Automorphic Functions and Fermat's Last Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 27 pages

[42] viXra:1012.0007 [pdf] replaced on 9 Jan 2011

Automorphic Functions and Fermat's Last Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 27 pages

[41] viXra:1012.0007 [pdf] replaced on 15 Dec 2010

Automorphic Functions and Fermat's Last Theorem(1)

Authors: Chun-Xuan Jiang
Comments: 7 pages

[40] viXra:1011.0077 [pdf] replaced on 15 Jan 2011

On "Discovering and Proving that π is Irrational"

Authors: Li Zhou
Comments: 7 pages.

[39] viXra:1010.0017 [pdf] replaced on 16 Dec 2010

Resolution of Riemann Hypothesis

Authors: Pankaj Mani
Comments: 5 pages

The Riemann hypothesis is proved to be true which states that all the non-trivial zeros of Riemann zeta function lie along the line R(z)=1/2 for 0<R(z)<1. The work done here clarifies that there is no need to find out the non-trivial zeros of the Riemann zeta function to prove the Riemann hypothesis true as the Riemann hypothesis must be true for the functional equation satisfied by the zeta function to exist itself structurally in mathematics.
Category: Number Theory

[38] viXra:1010.0006 [pdf] replaced on 17 Oct 2010

Power Structures in Finite Fields and the Riemann Hypothesis

Authors: Alessandro Dallari
Comments: 46 pages

Some tools are discussed, in order to build power structures of primitive roots in finite fields for any order q^k; relations between distinct roots are deduced from m- and shift-and-add- sequences. Some heuristic computational techniques, where information in a m- sequence is built from below, are proposed. Full settlement is finally viewed in a physical scenario, where a path leading to the Riemann Hypothesis can be enlighted.
Category: Number Theory

[37] viXra:1009.0054 [pdf] replaced on 17 Nov 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 6 pages

[36] viXra:1009.0054 [pdf] replaced on 16 Nov 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

[35] viXra:1009.0054 [pdf] replaced on 14 Nov 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

[34] viXra:1009.0054 [pdf] replaced on 13 Nov 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

[33] viXra:1009.0054 [pdf] replaced on 9 Nov 2010

An Elementary Proof of Catalan-Mihailescu Theorem

Authors: Jamel Ghanouchi
Comments: 5 pages

[32] viXra:1009.0053 [pdf] replaced on 17 Nov 2010

An Approach of Fermat-Catalan Equations

Authors: Jamel Ghanouchi
Comments: 24 pages

We begin with Beal equation (or Fermat-Catalan) U^c+2 = X^a+2 + Y^b+2 and establish two equivalent equations. We generalize the approach to all Fermat-Catalan equations which allows us to relate the problem to Matyasevich theorem. Our approach will lead us to propose a new conjecture concerning Fermat-Catalan equations. ( MSC=11D04) Keywords : Fermat-Catalan ; Diophantine equations ; Analysis ; Series ; Fourier series ; Conjecture.
Category: Number Theory

[31] viXra:1008.0022 [pdf] replaced on 29 Nov 2011

A Short Proof of Fermat's Last Theorem

Authors: Morgan D. Rosenberg
Comments: 11 pages

[30] viXra:1008.0001 [pdf] replaced on 20 Nov 2010

The Theory About Infinity of Simple Numbers-Twins

Authors: Valery Demidovich
Comments: v2 is 29 Pages in Russian, v3 is 28 pages in English

[29] viXra:1008.0001 [pdf] replaced on 19 Nov 2010

The Theory About Infinity of Simple Numbers-Twins

Authors: Valery Demidovich
Comments: 29 Pages. Russian version

[28] viXra:1005.0102 [pdf] replaced on 19 Jun 2010

The New Prime Theorem (45)-(70)

Authors: Chun-Xuan Jiang
Comments: 33 pages

Using Jiang function we prove that the new prime theorems (45)-(70) contain infinitely many prime solutions and no prime solutions.
Category: Number Theory

[27] viXra:1004.0126 [pdf] replaced on 4 May 2010

A Fifth Smarandache Friendly Prime Pair

Authors: Philip Gibbs
Comments: 4 pages

A Smarandache friendly prime pair (SFPP) is a pair of prime numbers (p,q), p < q, such that the product pq is equal to the sum of all primes from p to q inclusive. Previously four such pairs were known: (2,5), (3,13), (5,31) and (7,53). Now a fifth one is found by a brute force computer search. A heuristic approximation can be to estimate the expected number of SFPPs in a given interval. The result suggests that the probability of further pairs existing is about 0.07.
Category: Number Theory

[26] viXra:1004.0126 [pdf] replaced on 2 May 2010

A Fifth Smarandache Friendly Prime Pair

Authors: Philip Gibbs
Comments: 3 pages

[25] viXra:1004.0126 [pdf] replaced on 30 Apr 2010

A Fifth Smarandache Friendly Prime Pair

Authors: Philip Gibbs
Comments: 3 pages

[24] viXra:1004.0115 [pdf] replaced on 14 Jun 2010

Corrections to the wu-Sprung Potential for the Riemann Zeros and a New Hamiltonian Whose Energies Are the Prime Numbers

Authors: Jose Javier Garcia Moreta
Comments: 10 pages

[23] viXra:1004.0115 [pdf] replaced on 18 May 2010

Corrections to the wu-Sprung Potential for the Riemann Zeros and a New Hamiltonian Whose Energies Are the Prime Numbers

Authors: Jose Javier Garcia Moreta
Comments: 9 pages

[22] viXra:1004.0027 [pdf] replaced on 9 Jul 2011

Foundations of Santilli's Isonumber Theory

Authors: Chun-Xuan Jiang
Comments: 413 pages

[21] viXra:1003.0235 [pdf] replaced on 26 May 2010

Justification of the Zeta Regularization Procedure for the Integrals ∫x^m-Sdx

Authors: Jose Javier Garcia Moreta
Comments: 15 pages

[20] viXra:1003.0235 [pdf] replaced on 5 May 2010

Justification of the Zeta Regularization Procedure for the Integrals ∫x^m-Sdx

Authors: Jose Javier Garcia Moreta
Comments: 14 pages

[19] viXra:1003.0235 [pdf] replaced on 14 Apr 2010

Justification of the Zeta Regularization Procedure for the Integrals ∫x^m-Sdx

Authors: Jose Javier Garcia Moreta
Comments: 12 pages

[18] 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

[17] 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. [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

[16] viXra:1003.0179 [pdf] replaced on 2 Apr 2010

Prime Sieve Using Multiplication Operation Table

Authors: Jongsoo Park
Comments: 30 pages

[15] viXra:1003.0179 [pdf] replaced on 16 Mar 2010

Specific Composite Numbers Consist of Infinitely Many Arithmetic Progressions.

Authors: Jongsoo Park
Comments: 30 pages

[14] 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

[13] viXra:1003.0089 [pdf] replaced on 11 Mar 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: 40 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

[12] 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

[11] viXra:1003.0004 [pdf] replaced on 5 Mar 2010

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

Authors: Young-Mook Kang
Comments: 5 pages, Submitted to annals of mathematics

[10] viXra:1001.0042 [pdf] replaced on 28 Jun 2010

Zeta Regularization Applied to the Problem of Riemann Hypothesis and the Calculation of Divergent Integrals

Authors: Jose Javier Garcia Moreta
Comments: 18 Pages.

[9] viXra:1001.0038 [pdf] replaced on 7 Mar 2010

A Note on the Mellin Convolution of Functions and Its Relation to Riesz Criterion and Riemann Hypothesis

Authors: Jose Javier Garcia Moreta
Comments: 8 Pages.

[8] viXra:1001.0038 [pdf] replaced on 8 Feb 2010

A Note on the Mellin Convolution of Functions and Its Relation to Riesz Criterion and Riemann Hypothesis

Authors: Jose Javier Garcia Moreta
Comments: 7 Pages.

[7] viXra:0912.0043 [pdf] replaced on 21 Dec 2009

Imanol's Numbers

Authors: Imanol Pérez
Comments: 2 Pages.

Imanol's numbers are those that the sum of their digits is 2, 3, 5, 6, 8 or 9.
Category: Number Theory

[6] viXra:0911.0002 [pdf] replaced on 22 Nov 2009

A Perfect Number of Odd Numbers Doesn't Exist.

Authors: Kazuya Kawai
Comments: 2 pages

The mersenne prime number exists in infinity.
Category: Number Theory

[5] viXra:0911.0002 [pdf] replaced on 12 Nov 2009

A Perfect Number of Odd Numbers Doesn't Exist.

Authors: Kazuya Kawai
Comments: 2 pages

The mersenne prime number exists in infinity.
Category: Number Theory

[4] viXra:0911.0002 [pdf] replaced on 5 Nov 2009

Infinity of Mersenne Prime Number

Authors: Kazuya Kawai
Comments: 2 pages

The mersenne prime number exists in infinity.
Category: Number Theory

[3] viXra:0910.0021 [pdf] replaced on 30 May 2009

Removed

Authors:
Comments: 2 Pages

removed 0910.0021
Category: Number Theory

[2] viXra:0908.0091 [pdf] replaced on 25 Aug 2009

Polynomials with Rational Roots that Differ by a Non-Zero Constant

Authors: Philip Gibbs
Comments: 6 pages

[1] viXra:0812.0004 [pdf] replaced on 29 Dec 2008

Jiang's Function J_n+1(ω) in Prime Distribution

Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org

Jiang's function J_n+1(ω) in prime distribution
Category: Number Theory

Number Theory

Recent Submissions

The Largest Number Ever: a New Hierarchy of Super-Hyperoperators

Prime Distribution in Pythagorean Triples(1)(the Greatest Problem in Mathematics)

On Equivalence Between Zeta and R-Sequence

On P2^n Blocker Conjecture and R2 Function

The New Prime Theorems(1391)-(1440)

The New Prime Theorems 1341-1390

The New Prime Theorems (1291)-(1340)

The New Prime Theorems (1241)-(1290)

On a Strengthened Hardy-Hilbert's Type Inequality

Two Proofs of Catalan-Mihailescu Theorem

The New Prime Theorems (1191)-(1240)

Gauge Transformations and the Riemann Hypothesis

Relationship Between Irrational Constants Phi and e (Including New Equations, Possible Implications)

On the Sums (See Paper) and Their Relation to the Riemann Hypothesis and the Riemannweil Formula

Expression for the Mathematical Constant e

About Catalan-Mihailescu Theorem

Research on Some Smarandache Problems (Vol. 7)

A Resolution of Catalan Equation

About the Equation A/n = 1/x + 1/y + 1/z

The Algorithms of the Real Cube Root, the Positive Fourth Root, the Real ?fth Root and the Real Seventh Root of a Positive Number

Smarandache Deconstructive Sequence

Numeros Felizes e Sucessoes de Smarandache: Digressoes Com O Maple

Smarandache Sequence of Happy Cube Numbers

Pseudo-Smarandache Functions of First and Second Kind

Formula Per Trovare Numeri Primi

Fermat's Last Theorem, Goldbach's Conjecture and Riemann Hypothesis

Riemann Hypothesis Resolution Using Nash Equilibrium to Find the Best Location of Non-Trivial Zeros and Discovering the Locality-at-a Distance (Bell's Theorem) in Mathematical Field

The Proofs of Binary Goldbach's Theorem Using Only Partial Primes

Generalized Fermat's Last Theorem (3)

Fermat-Catalan Equations (1)

Brocard`s Problem. Variants of Brocard`s Problem

Generalized Fermat's Last Theorem

Fermat's Last Theorem (1)

A Generalisation of Fermat-Catalan Conjecture

Collatz Problem and Conjecture. a Generalization of the Problem

Patterns Related to the Smarandache Circular Sequence Primality Problem

Generalized Fermat's Last Theorem Rn = ...

Fermat Last Theorem Controversy (6)

Jiang and Wiles Who Has First Proved Fermat Last Theorem (3)

Jiang and Wiles Who Has First Proved Fermat Last Theorem (6)

Jiang and Wiles Who Has First Proved Fermat Last Theorem (5)

Jiang and Wiles Who Has First Proved Fermat Last Theorem (4)

Jiang and Wiles Who Has First Proved Fermat Last Theorem (2)

Fermat Last Theorem Controversy (2)

Une Généralisation de la Conjecture de Fermat-Catalan

The Powers of π Are Irrational

The Diophantine Equations A2 ± M B2 = Cn, A3 ± M B3 = D2 and Y14 ± M Y24 = R2

Jiang and Wiles Who Has First Proved Fermat Last Theorem (1)

An Analytical Approach to Polyominoes and a Solution to the Goldbach Conjecture

Jiang and Wiles Proofs on Fermat Last Theorem (4)

Distribution of Prime Numbers,twin Primes and Goldbach Conjecture

Jiang and Wiles Proofs on Fermat Last Theorem (3)

Jiang and Wiles Proofs on Fermat Last Theorem(2)

On Prime Factors in Old and New Sequences of Integers

Fermat Last Theorem and Riemann Hypothesis (6)

Fermat Last Theorem and Riemann Hypothesis (5)

Fermat Last Theorem and Riemann Hypothesis (4)

Generation of DNA Codes & the Hexagrams of I Ching from the Principle of Existence

Jiang and Wiles Proofs on Fermat Last Theorem(1)

Remarks on the Function η(n)

New Progress on Smarandache Problems Research

The K-Number Sieve and K-Inclusion-Exclusion Formula, Principle and Harvest

The New Prime Theorems (1041)-(1090)

The New Prime Theorems (991)-(1040)

Counter to Zhou's Criticism of Jones' Proof of the Irrationality of π and π2

The New Prime Theorems (941)-(990)

The New Prime Theorems (841)-(890)

A Treaty of Symmetric Function Part V Using Sum of Power for Arbitrary Arithmetic Progression for Studies of Prime Numbers that Coexist Within the Equation Formed Through a New Conjecture of Symmetric Function Rule of Division

A Treaty of Symmetric Function Part IV Using Sums of Power for Arbitrary Arithmetic Progression to Find an Approximation for Sum of Power for Non-Integers R-th Power and Expressing Riemann's Zeta Function Into Symmetric Sums of Power Form.

A Treaty of Symmetric Function Part III an Approach in Deriving General Formulation for Alternating Sums of Power for an Arbitrary Arithmetic Progression.

A Treaty of Symmetric Function Part II Sums of Power for an Arbitrary Arithmetic Progression for Real Power-P

A Treaty of Symmetric Function Part 1 Sums of Power

Automorphic Function and Fermat's Last Theorem (6)

Automorphic Function and Fermat's Last Theorem (5)

Aotomorphic Functions and Fermat's Last Theorem (4)

Automorphic Function and Fermat's Last Theorem (3) (Fermat's Proof of FLT)

Automorphic Function and Fermat's Last Theorem(2)

Automorphic Functions and Fermat's Last Theorem(1)

Generalized Fermat's Last Theorem Rⁿ = ...

The Diophantine Equations A² ± M B² = Cⁿ, A³ ± M B³ = D² and Y₁⁴ ± M Y₂⁴ = R²

Counter to Zhou's Criticism of Jones' Proof of the Irrationality of π and π²

Goldbach' Conjecture (5): When I=1~r, the P and N Are Incongruent Modulo P_i, the P is Goldbach' Primes

Goldbach' Conjecture (2): When the P is Congruent to N Modulo P_i, the P is not Goldbach' Primes

A Method to Solve the Diophantine Equation a X²-B Y² + C = 0