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[201] viXra:1009.0004 [pdf] submitted on 2 Sep 2010
Authors: Kunikazu Tanaka
Comments:
21 pages
Showing how to derive new expressions of generating prime numbers to demonstrate the Goldbach's Conjecture
[200] viXra:1008.0089 [pdf] submitted on 30 Aug 2010
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. πk(N0,2) ≥ 1. This is the Book theorem.
[199] viXra:1008.0088 [pdf] submitted on 31 Aug 2010
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).
[198] viXra:1008.0087 [pdf] submitted on 30 Aug 2010
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. πk(N0,2) ≥ 1. This is the Book theorem.
[197] viXra:1008.0086 [pdf] submitted on 30 Aug 2010
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. πk(N0,2) ≥ 1. This is the Book theorem.
[196] viXra:1008.0082 [pdf] submitted on 13 Mar 2010
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.
[195] viXra:1008.0080 [pdf] submitted on 27 Aug 2010
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.
[194] viXra:1008.0069 [pdf] submitted on 25 Aug 2010
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.
[193] viXra:1008.0064 [pdf] submitted on 23 Aug 2010
Authors: Tong Xin Ping
Comments: 3 pages, In Chinese
This upper bound estimation prevailed over upper bound estimation of Chen Jing Run
[192] viXra:1008.0062 [pdf] submitted on 22 Aug 2010
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"
[191] viXra:1008.0061 [pdf] submitted on 22 Aug 2010
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 Σn 1/Z(n)α is convergent for any α > 1.
[190] viXra:1008.0054 [pdf] submitted on 20 Aug 2010
Authors: Tong Xin Ping
Comments: 4 pages, In Chinese
According to five assumptions, get five proofs
[189] viXra:1008.0041 [pdf] submitted on 13 Aug 2010
Authors: Florentin Smarandache
Comments: 175 Pages. In French
Problems with and without... problems!
[188] viXra:1008.0036 [pdf] submitted on 12 Aug 2010
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.
[187] viXra:1008.0022 [pdf] submitted on 8 Aug 2010
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.
[186] viXra:1008.0021 [pdf] submitted on 8 Aug 2010
Authors: Tong Xin Ping
Comments: 2 pages, In Chinese
Don't confuse quantitative change and qualitative change.
[185] viXra:1008.0006 [pdf] submitted on 4 Aug 2010
Authors: Tong Xin Ping
Comments: 2 Pages. In Chinese
The method of the quantitative change can not solve the problem of the qualitative change.
[184] viXra:1008.0001 [pdf] submitted on 1 Aug 2010
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.
[183] viXra:1007.0049 [pdf] submitted on 28 Jul 2010
Authors: Tong Xin Ping
Comments: 4 pages. In Chinese
By the Chinese Remainder Theorem, we can obtain Goldbach' Primes
[182] viXra:1007.0048 [pdf] submitted on 28 Jul 2010
Authors: Tong Xin Ping
Comments: 2 pages. In Chinese
When i=1~r, the p and N are incongruent modulo pi, The p is Goldbach' Primes
[181] viXra:1007.0046 [pdf] submitted on 27 Jul 2010
Authors: Tong Xin Ping
Comments: 3 pages. In Chinese
Use the inclusion-exclusion to show that the expression of the number of Goldbach' Primes.
[180] viXra:1007.0045 [pdf] submitted on 27 Jul 2010
Authors: Tong Xin Ping
Comments: 1 pages. In Chinese
By Eratosthenes' sieve method, we can obtain Goldbach' Primes.
[179] viXra:1007.0037 [pdf] submitted on 24 Jul 2010
Authors: Tong Xin Ping
Comments: 2 pages.
When the p is congruent to N modulo pi, the p is not Goldbach' Primes.
[178] viXra:1007.0036 [pdf] submitted on 24 Jul 2010
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.
[177] viXra:1007.0025 [pdf] submitted on 17 Jul 2010
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.
[176] viXra:1007.0021 [pdf] submitted on 10 Jul 2010
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.
[175] viXra:1007.0015 [pdf] submitted on 13 Mar 2010
Authors: Florentin Smarandache
Comments: 3 pages
We define a class of sequences {an} by a1 = a and an+1 = P(an), 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.
[174] viXra:1007.0013 [pdf] submitted on 10 Jul 2010
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.
[173] viXra:1007.0002 [pdf] submitted on 2 Jul 2010
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.
[172] viXra:1006.0060 [pdf] submitted on 13 Mar 2010
Authors: Florentin Smarandache
Comments:
10 pages.
We consider the equation ...
[171] viXra:1006.0048 [pdf] submitted on 19 Jun 2010
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.
[170] viXra:1006.0047 [pdf] submitted on 19 Jun 2010
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.
[169] viXra:1006.0020 [pdf] submitted on 11 Jun 2010
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.
[168] viXra:1006.0016 [pdf] submitted on 11 Mar 2010
Authors: Felice Russo
Comments: 3 pages
In this paper some properties of the Smarandache double factorial function have been analyzed.
[167] viXra:1006.0014 [pdf] submitted on 11 Mar 2010
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.
[166] viXra:1006.0001 [pdf] submitted on 2 Jun 2010
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.
[165] viXra:1005.0109 [pdf] submitted on 11 Mar 2010
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.
[164] viXra:1005.0107 [pdf] submitted on 11 Mar 2010
Authors: Yi Yuan, Zhang Wenpeng
Comments: 3 pages
see paper for abstract
[163] viXra:1005.0106 [pdf] submitted on 11 Mar 2010
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.
[162] viXra:1005.0105 [pdf] submitted on 11 Mar 2010
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) .
[161] viXra:1005.0102 [pdf] submitted on 29 May 2010
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.
[160] viXra:1005.0096 [pdf] submitted on 24 May 2010
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).
[159] viXra:1005.0092 [pdf] submitted on 11 Mar 2010
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.
[158] viXra:1005.0088 [pdf] submitted on 21 May 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function J2(ω) we prove that jPn + 9 - j contain infinitely many prime solutions.
[157] viXra:1005.0087 [pdf] submitted on 21 May 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that jP8 + k - j contain infinitely many prime solutions.
[156] viXra:1005.0086 [pdf] submitted on 21 May 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that jP7 + k - j contain infinitely many prime solutions.
[155] viXra:1005.0085 [pdf] submitted on 21 May 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that jP6 + k - j contain infinitely many prime solutions.
[154] viXra:1005.0084 [pdf] submitted on 21 May 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that jP5 + k - j contain infinitely many prime solutions.
[153] viXra:1005.0083 [pdf] submitted on 21 May 2010
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.
[152] viXra:1005.0067 [pdf] submitted on 11 Mar 2010
Authors: Felice Russo
Comments:
5 pages.
The Smarandache P and S persistence of a prime
[151] viXra:1005.0064 [pdf] submitted on 15 May 2010
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)
[150] viXra:1005.0058 [pdf] submitted on 11 Mar 2010
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.
[149] viXra:1005.0054 [pdf] submitted on 11 Mar 2010
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.
[148] viXra:1005.0049 [pdf] submitted on 11 Mar 2010
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.
[147] viXra:1005.0047 [pdf] submitted on 11 Mar 2010
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.
[146] viXra:1005.0042 [pdf] submitted on 11 May 2010
Authors: Chun-Xuan Jiang
Comments: 2 Pages
Using Jiang function we prove for any there are infinitely many primes P such that each of jPP0 + j+1 is a prime.
[145] viXra:1005.0041 [pdf] submitted on 11 May 2010
Authors: Chun-Xuan Jiang
Comments: 2 Pages
Using Jiang function we prove for any there are infinitely many primes P such that each of PP0 + 4n is a prime.
[144] viXra:1005.0040 [pdf] submitted on 11 May 2010
Authors: Chun-Xuan Jiang
Comments: 2 Pages
Using Jiang function we prove for any there are infinitely many primes kPsuch that each of PP0 + (2j)2 is a prime.
[143] viXra:1005.0039 [pdf] submitted on 11 May 2010
Authors: Chun-Xuan Jiang
Comments: 2 Pages
Using Jiang function we prove for any there are infinitely many primes kPsuch that each of PP0 + j(j+1) is a prime.
[142] viXra:1005.0038 [pdf] submitted on 11 May 2010
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.
[141] viXra:1005.0037 [pdf] submitted on 11 May 2010
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.
[140] viXra:1005.0036 [pdf] submitted on 11 May 2010
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.
[139] viXra:1005.0035 [pdf] submitted on 11 May 2010
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.
[138] viXra:1005.0032 [pdf] submitted on 9 May 2010
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.
[137] viXra:1005.0031 [pdf] submitted on 9 May 2010
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.
[136] viXra:1005.0030 [pdf] submitted on 9 May 2010
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.
[135] viXra:1005.0029 [pdf] submitted on 9 May 2010
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.
[134] viXra:1005.0028 [pdf] submitted on 9 May 2010
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.
[133] viXra:1005.0027 [pdf] submitted on 9 May 2010
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.
[132] viXra:1005.0025 [pdf] submitted on 10 May 2010
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.
[131] viXra:1005.0023 [pdf] submitted on 11 Mar 2010
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.
[130] viXra:1005.0017 [pdf] submitted on 5 May 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 2 pages
About an Identity and its Applications
[129] viXra:1005.0008 [pdf] submitted on 2 May 2010
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
[128] viXra:1004.0140 [pdf] submitted on 10 Mar 2010
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.
[127] viXra:1004.0135 [pdf] submitted on 30 Apr 2010
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.
[126] viXra:1004.0134 [pdf] submitted on 30 Apr 2010
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.
[125] viXra:1004.0133 [pdf] submitted on 30 Apr 2010
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.
[124] viXra:1004.0132 [pdf] submitted on 30 Apr 2010
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.
[123] viXra:1004.0131 [pdf] submitted on 30 Apr 2010
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.
[122] viXra:1004.0126 [pdf] submitted on 28 Apr 2010
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.
[121] viXra:1004.0125 [pdf] submitted on 10 Mar 2010
Authors: Felice Russo
Comments:
3 pages.
In this paper a question posed in [1] and concerning the Smarandache friendly prime pairs is analysed.
[120] viXra:1004.0123 [pdf] submitted on 27 Apr 2010
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)
[119] viXra:1004.0122 [pdf] submitted on 27 Apr 2010
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).
[118] viXra:1004.0119 [pdf] submitted on 24 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove and P1 = P9 ± m and P1 = (2P)9 ± n
[117] viXra:1004.0118 [pdf] submitted on 24 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove and P1 = PP0 ± m and P1 = (2P)p0 ± n
[116] viXra:1004.0117 [pdf] submitted on 24 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove and P1 = P5 ± m and P1 = (2P)5 ± n
[115] viXra:1004.0116 [pdf] submitted on 24 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that 1P = P ± m and 1 P = 2P ± n have infinitely many
[114] viXra:1004.0115 [pdf] submitted on 23 Apr 2010
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 loge (x) = ln(x) = log(x) for the logarithm , also unles we specify Σγ h(γ) means that we sum over ALL the imaginary parts of the nontrivial zero on both the upper and lower complex plane.
[113] viXra:1004.0111 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove Hardy-Littlewood conjecture P: m2 +1 and m2 + 3 [4].
[112] viXra:1004.0110 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove Hardy-Littlewood conjecture N: x3 + y3 + z3 [4].
[111] viXra:1004.0109 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove Hardy-Littlewood conjecture M: x3 + y3 + k [4].
[110] viXra:1004.0108 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove Hardy-Littlewood conjecture K: x3 + k [4].
[109] viXra:1004.0107 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove Hardy-Littlewood conjecture F: am2 + bm+ c [4].
[108] viXra:1004.0106 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove Hardy-Littlewood conjecture B: P, P + k [4].
[107] viXra:1004.0105 [pdf] submitted on 20 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove binary Goldbach conjecture and N = P1 + ... + Pn [4]
[106] viXra:1004.0104 [pdf] submitted on 20 Apr 2010
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].
[105] viXra:1004.0088 [pdf] submitted on 18 Apr 2010
Authors: Tong Xin Ping
Comments: 6 Pages, In Chinese
(see paper)
[104] viXra:1004.0087 [pdf] submitted on 10 Mar 2010
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 .
[103] viXra:1004.0071 [pdf] submitted on 10 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove that such that (see paper) has infinitely many prime solutions.
[102] viXra:1004.0070 [pdf] submitted on 10 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove Hardy-Littlewood conjecture E : x2 + 1
[101] viXra:1004.0069 [pdf] submitted on 10 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove that such that Pn = 2 P1P2 ... Pn-1 has infinitely many prime solutions.
[100] viXra:1004.0068 [pdf] submitted on 10 Apr 2010
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.
[99] viXra:1004.0067 [pdf] submitted on 10 Apr 2010
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.
[98] viXra:1004.0066 [pdf] submitted on 10 Apr 2010
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.
[97] viXra:1004.0060 [pdf] submitted on 8 Apr 2010
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.
[96] viXra:1004.0059 [pdf] submitted on 8 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that 3 x a3 ± 1 has infinitely many prime solutions
[95] viXra:1004.0058 [pdf] submitted on 8 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that 2 x a2 ± 1 has infinitely many prime solutions
[94] viXra:1004.0045 [pdf] submitted on 6 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove the finite Mersenne primes and the finite repunits primes.
[93] viXra:1004.0044 [pdf] submitted on 6 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove the finite fermat primes.
[92] viXra:1004.0043 [pdf] submitted on 6 Apr 2010
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.
[91] viXra:1004.0042 [pdf] submitted on 6 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove prime theorem: P2 = aP1 + b, Polignac theorem and Goldbach theorem.
[90] viXra:1004.0041 [pdf] submitted on 8 Mar 2010
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)
[89] viXra:1004.0040 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache
Comments: 2 pages
Back and Forth Factorials
[88] viXra:1004.0038 [pdf] submitted on 8 Mar 2010
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.
[87] viXra:1004.0034 [pdf] submitted on 4 Apr 2010
Authors: Chun-Xuan Jiang
Comments: 1 page
Using Jiang function we prove that x6 + 1091 has no prime solutions.
[86] viXra:1004.0033 [pdf] submitted on 4 Apr 2010
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.
[85] viXra:1004.0032 [pdf] submitted on 4 Apr 2010
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.
[84] viXra:1004.0031 [pdf] submitted on 4 Apr 2010
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.
[83] viXra:1004.0030 [pdf] submitted on 4 Apr 2010
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.
[82] viXra:1004.0029 [pdf] submitted on 4 Apr 2010
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.
[81] viXra:1004.0028 [pdf] submitted on 5 Apr 2010
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 Jn(ω) that can replace Riemann's ζ(s) function in view of its proved features: if Jn(ω) ≠ 0, then the function has infinitely many prime solutions; and if Jn(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.
[80] viXra:1004.0027 [pdf] submitted on 4 Apr 2010
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.
[79] viXra:1004.0020 [pdf] submitted on 8 Mar 2010
[78] viXra:1003.0274 [pdf] submitted on 31 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that there exist infinitely many primes P such that P1 and P2 are all prime.
[77] viXra:1003.0273 [pdf] submitted on 31 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove that there exist infinitely many primes P1 such that a P1 + b is prime.
[76] viXra:1003.0271 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache
Comments: 3 pages
On a Problem with Primes.
[75] viXra:1003.0264 [pdf] submitted on 30 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that for every positive integer k there exist infinitely many primes P such that each of P + 4n is prime.
[74] viXra:1003.0263 [pdf] submitted on 30 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that for every positive integer k there exist infinitely many primes P1 and P2 such that each of 1 2 jP1 + (j + 1)P2 is prime.
[73] viXra:1003.0262 [pdf] submitted on 30 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that for every positive integer k there exist infinitely many primes P such that each of P + (2j)2 is prime.
[72] viXra:1003.0261 [pdf] submitted on 30 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 1 pages
Using Jiang function we prove that for every positive integer k there exist infinitely many primes P such that each of jP + j +1 is prime.
[71] viXra:1003.0260 [pdf] submitted on 30 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 3 pages
Using Jiang function we prove that for every positive integer k there exist infinitely many primes P such that each of P + j(j + 1) is prime
[70] viXra:1003.0258 [pdf] submitted on 28 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jinag funciton we prove that there exist infinitely many primes 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.
[69] viXra:1003.0235 [pdf] submitted on 23 Mar 2010
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 ∫xm-sdx via the zeta regularization of the divergent series Σxm-sdx and the zeta function ζ(m - s)
[68] viXra:1003.0234 [pdf] submitted on 23 Mar 2010
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.
[67] viXra:1003.0233 [pdf] submitted on 7 Mar 2010
Authors: Florentin Smarandache
Comments: 141 pages
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein.
[66] viXra:1003.0230 [pdf] submitted on 7 Mar 2010
Authors: Sebastián Martín Ruiz
Comments: 25 pages
The Smarandache function is defined as follows: S(n)= the smallest positive integer such that S(n)! is divisible by n. [1] In this article we are going to see that the value this function takes when n is a perfect number of the form n = 2k - 1.(2k - 1) , p = 2k - 1 being a prime number.
[65] viXra:1003.0228 [pdf] submitted on 7 Mar 2010
Authors: Amarnath Murthy, Charles Ashbacher
Comments: 219 pages
This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was to remove the duplications, organize the material based on topic and clean up some of the most obvious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy.
[64] viXra:1003.0225 [pdf] submitted on 7 Mar 2010
Authors: József Sándor
Comments: 302 pages
This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into five chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry
[63] viXra:1003.0220 [pdf] submitted on 7 Mar 2010
Authors: Charles Ashbacher
Comments: 80 pages
In writing a book, one encounters and overcomes many obstacles. Not the least of which is the occasional case of writer's block. This is especially true in mathematics where sometimes the answer is currently and may for all time be unknown. There is nothing worse than writing yourself into a corner where your only exit is to build a door by solving unsolved problems. In any case, it is my hope that you will read this volume and come away thinking that I have overcome enough of those obstacles to make the book worthwhile. As always, your comments and criticisms are welcome. Feel free to contact me using any of the addresses listed below, although e-mail is the preferred method.
[62] viXra:1003.0219 [pdf] submitted on 7 Mar 2010
Authors: Charles Ashbacher
Comments: 135 pages
This is the fifth book that I have written that expands on the ideas of Florentin Smarandache. In addition, I have edited two others that also deal with the areas of mathematics under the Smarandache Notions umbrella. All of this is a credit to the breadth and depth of his mathematical achievement. Therefore, I once again must commend and thank him for providing so much material to work with. I also would like to thank J. McGray for her encouragement and patience as I struggled to make this book a reality. The material cited in this book can be found at the website http://www.gallup.unm.edu/~smarandache/. The deepest thanks go to my mother Paula Ashbacher, who encouraged me to play sports, but in the off chance that I would never learn to hit the curve ball, also insisted that I read books. This proved to be a wise career strategy. Finally, I would like to express my deep love for Kathy Brogla, my partner/soul mate/best friend. So pretty and vivacious, she makes life fun, exciting and a joy to experience every single day. She is a remarkable woman and I am so blessed to have her in my life. Kathy is also the creator of the image on the front cover.
[61] viXra:1003.0217 [pdf] submitted on 7 Mar 2010
Authors: Henry Ibstedt
Comments: 97 pages
This book consists of a selection of papers most of which were produced during the period 1999-2002. They have been inspired by questions raised in recent articles in current Mathematics journals and in Florentin Smarandache's wellknown publication Only Problems, Not Solutions.
[60] viXra:1003.0216 [pdf] submitted on 7 Mar 2010
Authors: C. Dumitrescu, V. Seleacu
Comments: 137 pages
The function named in the title of this book is originated from the exiled Romanian mathematician Florentin Smarandache.
[59] viXra:1003.0211 [pdf] submitted on 18 Mar 2010
Authors: Tong Xin Ping
Comments: 4 Pages, In Chinese
We have sieve method formula of π(N) and sieve method formula of r2(N). By these sieve method formula, we can obtain (see paper for equation)
[58] viXra:1003.0199 [pdf] submitted on 6 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 206 pages
Historically a code refers to a cryptosystem that deals with linguistic units: words, phrases etc. We do not discuss such codes in this book. Here codes are message carriers or information storages or information transmitters which in time of need should not be decoded or read by an enemy or an intruder. When we use very abstract mathematics in using a specific code, it is difficult for non-mathematicians to make use of it. At the same time, one cannot compromise with the capacity of the codes. So the authors in this book have introduced several classes of codes which are explained very non-technically so that a strong foundation in higher mathematics is not needed. The authors also give an easy method to detect and correct errors that occur during transmission. Further some of the codes are so constructed to mislead the intruder. False n-codes, whole n-codes can serve this purpose.
[57] viXra:1003.0198 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 3 pages
TSix conjectures on pairs of consecutive primes are listed below together with examples in each case.
[56] viXra:1003.0189 [pdf] submitted on 16 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 13 pages
Santilli's prime chains: (see paper for equations) There exist infinitely many primes such that are primes for arbitrary length . It is the Book proof. This is a generalization of Euclid-Euler proof for the existence of infinitely many primes. Therefore Euclid-Euler-Jiang theorem in the distribution of primes is advanced. It is the Book theorem.
[55] viXra:1003.0188 [pdf] submitted on 16 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 5 pages
Using Jiang's function we prove that there are infinitely many primes such that 3P-2 and 3P+2 are primes.
[54] viXra:1003.0186 [pdf] submitted on 6 Mar 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 8 pages
In this paper we present some new inequalities relative to integer and functional parts.
[53] viXra:1003.0180 [pdf] submitted on 6 Mar 2010
Authors: C. Dumitrescu, N. Vîrlan, Şt. Zamfir, E. Rădescu, N. Rădescu, Florentin Smarandache
Comments: 15 pages
In this paper we extended the Smarandache function from the set N* of positive integers to the set Q of rational numbers. Using the inversion formula, this function is also regarded as a generating function. We put in evidence a procedure to construct a (numerical) function starting from a given function in two particular cases. Also connections between the Smarandache function and Euler's totient function as with Riemann's zeta function are established.
[52] viXra:1003.0178 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
The utility of this article is that it establishes if the number of the natural solutions of a general linear equation is limited or not. We will show also a method of solving, using integer numbers, the equation ax - by = c (which represents a generalization of lemmas 1 and 2 of [4]), an example of solving a linear equation with 3 unknowns in N, and some considerations on solving, using natural numbers, equations with n unknowns.
[51] viXra:1003.0177 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
In this article we present four necessary and sufficient conditions for a natural number to be prime.
[50] viXra:1003.0170 [pdf] submitted on 14 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 18 pages
By using the arithmetic function J2n+1(ω) we prove that Diophantine equation (see paper) has infinitely many prime solutions.It is the Book proof. The J2n+1(ω) ushers in a new era in the prime numbers theory.
[49] viXra:1003.0163 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 7 pages
In the paragraphs which follow we will prove a result which replaces the theorem of Euler: "If (a,m) = 1, then aφ(m) = 1 (mod m)", for the case when a and m are not relatively primes.
[48] viXra:1003.0160 [pdf] submitted on 6 Mar 2010
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 (xn,yn), the general positive integer solution, by an original method. More, we generalize it for any Diophantine equation of second degree and with two unknowns.
[47] viXra:1003.0153 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 3 pages
In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.
[46] viXra:1003.0151 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 11 pages
(see paper)
[45] viXra:1003.0139 [pdf] submitted on 12 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 9 pages
Using Jiang function J2(ω) we prove gaps among products of m prime: d(x) = d(x + 1) = d(x + 5 - 3) = d(x + 7 - 3) = ... = d(x + Pn - 3) = m > 1 infinitely-often, where Pn denotes the n - th prime.
[44] viXra:1003.0122 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 6 pages
A Generalized Numeration Base is defined in this paper, and then particular cases are presented, such as Prime Base, Square Base, m-Power Base, Factorial Base, and operations in these bases.
[43] viXra:1003.0121 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 14 pages
Other new sequences are introduced in number theory, and for each one a general question: how many primes each sequence has.
[42] viXra:1003.0120 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 42 pages
New sequences in number theory are showed below with definitions, examples, solved or open questions and references for each case.
[41] viXra:1003.0118 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 16 pages
A collection of original sequences, open questions, and problems are mentioned below.
[40] viXra:1003.0112 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
In this article we will construct a family of expressions ε(n). For each element E(n) from ε(n), the convergence of the series Σ E(n) can be determined in accordance to the theorems of this article.
[39] viXra:1003.0111 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 8 pages
In this paper we define a function L which will allow us to (separately or simultaneously) generalize many theorems from Number Theory obtained by Wilson, Fermat, Euler, Gauss, Lagrange, Leibniz, Moser, and Sierpinski.
[38] viXra:1003.0107 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 9 pages
This article presents a necessary and sufficient theorem for N numbers, coprime two by two, to be prime simultaneously. It generalizes V. Popa's theorem [3], as well as I. Cucurezeanu's theorem ([1], p. 165), Clement's theorem, S. Patrizio's theorems [2], etc. Particularly, this General Theorem offers different characterizations for twin primes, for quadruple primes, etc.
[37] viXra:1003.0103 [pdf] submitted on 6 Mar 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 11 pages
In this paper we give a method, based on the characteristic function of a set, to solve some difficult problems of set theory found in undergraduate studies.
[36] viXra:1003.0102 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
On Carmichaël's conjecture
[35] viXra:1003.0095 [pdf] submitted on 6 Mar 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 3 pages
Many methods to compute the sum of the first n natural numbers of the same powers (see [4]) are well known. In this article we present a simple proof of the method from [3].
[34] viXra:1003.0093 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 5 pages
In this article we establish some properties regarding the solutions of a linear congruence, bases of solutions of a linear congruence, and the finding of other solutions starting from these bases. This article is a continuation of my article "On linear congruences".
[33] viXra:1003.0089 [pdf] submitted on 8 Mar 2010
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.
[32] viXra:1003.0087 [pdf] submitted on 8 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 7 pages, Dedicated to the 30-th anniversary of China reform and opening
We establish the Santilli's isomathematics based on the generalization of the modern mathematics. (more see paper)
[31] viXra:1003.0086 [pdf] submitted on 8 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 5 pages
In this paper we prove that it is sufficient to prove 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.
[30] viXra:1003.0085 [pdf] submitted on 8 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 2 pages
Using Jiang function we prove prime theorem: P2 = aP1 + b Polignac theorem and Goldbach theorem.
[29] viXra:1003.0084 [pdf] submitted on 8 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 4 pages
We find Blasius function to satisfy the boundary condition f'(∞) = 1 and obtain the approximate solutions of Blasius equation.
[28] viXra:1003.0069 [pdf] submitted on 6 Mar 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 2 pages
In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals.
[27] viXra:1003.0068 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 2 pages
In this note we present a method of solving this Diophantine equation, method which is different from Ljunggren's, Mordell's, and R.K.Guy's.
[26] viXra:1003.0067 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 9 pages
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences and we find the number of distinct solutions. Many examples of solving congruences are given.
[25] viXra:1003.0063 [pdf] submitted on 6 Mar 2010
Authors: Mihály Bencze, Florentin Smarandache
Comments: 3 pages
In this paper, we present some new inequalities for factorial sum.
[24] viXra:1003.0061 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 38 pages
Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and geometric progressions are exposed.
[23] viXra:1003.0004 [pdf] submitted on 4 Mar 2010
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(x1/2+ε) for ε > 0. Also Littlewood proved that "the RH is equivalent to the statement that limx → ∞ M(x)x-1/2-ε = 0, for every ε > 0".[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.
[22] viXra:1002.0024 [pdf] submitted on 14 Feb 2010
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.
[21] viXra:1001.0047 [pdf] submitted on 29 Jan 2010
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)
[20] viXra:1001.0042 [pdf] submitted on 27 Jan 2010
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)
[19] viXra:1001.0039 [pdf] submitted on 26 Jan 2010
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.
[18] viXra:1001.0038 [pdf] submitted on 26 Jan 2010
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.
[17] viXra:0912.0043 [pdf] submitted on 19 Dec 2009
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.
[16] viXra:0912.0040 [pdf] submitted on 18 Dec 2009
Authors: Imanol Pérez
Comments: 2 Pages. In Spanish
Expansion of (1/x+2/x.......+a/x)n
[15] viXra:0912.0030 [pdf] submitted on 12 Dec 2009
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 < 10106. 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.
[14] viXra:0911.0002 [pdf] submitted on 2 Nov 2009
Authors: Kazuya Kawai
Comments: 2 pages
The mersenne prime number exists in infinity.
[13] viXra:0910.0012 [pdf] submitted on 9 Oct 2009
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.
[12] viXra:0909.0034 [pdf] submitted on 14 Sep 2009
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 sn = 1/2 + iλn. 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 < Ψs | CT | Ψs > ≠ 0 where s are the complex eigenvalues of the scaling-like operators.
[11] viXra:0908.0098 [pdf] submitted on 26 Aug 2009
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 sn = 1/2 + iλn. 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.
[10] viXra:0908.0091 [pdf] submitted on 24 Aug 2009
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.
[9] viXra:0908.0079 [pdf] submitted on 21 Aug 2009
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 ρn (imaginary parts) of the zeta zeros in the critical line : sn = 1/2 + iρn A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (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.
[8] viXra:0908.0050 [pdf] submitted on 10 Aug 2009
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.
[7] viXra:0907.0024 [pdf] submitted on 20 Jul 2009
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.
[6] viXra:0904.0003 [pdf] submitted on 7 Apr 2009
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)Pk are all composites for arbitrarily long k, where P1, P2, ..., Pk are all primes. This result has no prior occurrence in the history of number theory.
[5] viXra:0904.0001 [pdf] submitted on 6 Apr 2009
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 < Pg+1 long arithmetic progressions, but the primes have no k > Pg+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.
[4] viXra:0901.0003 [pdf] submitted on 14 Jan 2009
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.
[3] viXra:0901.0002 [pdf] submitted on 3 Jan 2009
Authors: Tong Xin Ping
Comments: recovered from sciprint.org
N = pi + (N-pi) = p+ (N-p). If p is congruent to N modulo pi, Then (N-p) is a composite integer, When i = 1, 2,..., r, if p and N are incongruent modulo pi, 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).
[2] viXra:0812.0009 [pdf] submitted on 29 Dec 2008
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.
[1] viXra:0812.0004 [pdf] submitted on 9 Dec 2008
Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org
see paper
[26] viXra:1005.0102 [pdf] replaced on 19 Jun 2010
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.
[25] viXra:1004.0126 [pdf] replaced on 4 May 2010
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.
[24] viXra:1004.0126 [pdf] replaced on 2 May 2010
Authors: Philip Gibbs
Comments: 3 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.
[23] viXra:1004.0126 [pdf] replaced on 30 Apr 2010
Authors: Philip Gibbs
Comments: 3 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.
[22] viXra:1004.0115 [pdf] replaced on 14 Jun 2010
Authors: Jose Javier Garcia Moreta
Comments: 10 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 loge (x) = ln(x) = log(x) for the logarithm , also unles we specify Σγ h(γ) means that we sum over ALL the imaginary parts of the nontrivial zero on both the upper and lower complex plane.
[21] viXra:1004.0115 [pdf] replaced on 18 May 2010
Authors: Jose Javier Garcia Moreta
Comments: 9 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 loge (x) = ln(x) = log(x) for the logarithm , also unles we specify Σγ h(γ) means that we sum over ALL the imaginary parts of the nontrivial zero on both the upper and lower complex plane.
[20] viXra:1003.0235 [pdf] replaced on 26 May 2010
Authors: Jose Javier Garcia Moreta
Comments: 15 pages
In this paper we review and try to justify some results we gave before concerning the zeta regularization of integrals ∫xm-sdx via the zeta regularization of the divergent series Σxm-sdx and the zeta function ζ(m - s)
[19] viXra:1003.0235 [pdf] replaced on 5 May 2010
Authors: Jose Javier Garcia Moreta
Comments: 14 pages
In this paper we review and try to justify some results we gave before concerning the zeta regularization of integrals ∫xm-sdx via the zeta regularization of the divergent series Σxm-sdx and the zeta function ζ(m - s)
[18] viXra:1003.0235 [pdf] replaced on 14 Apr 2010
Authors: Jose Javier Garcia Moreta
Comments: 12 pages
In this paper we review and try to justify some results we gave before concerning the zeta regularization of integrals ∫xm-sdx via the zeta regularization of the divergent series Σxm-sdx and the zeta function ζ(m - s)
[17] viXra:1003.0234 [pdf] replaced on 26 Mar 2010
Authors: Chun-Xuan Jiang
Comments: 7 pages
Using Jiang function we prove Jiang prime -tuple theorem. We prove that the Hardy-Littlewood prime-tuple conjecture is false. Jiang prime -tuple theorem can replace the Hardy-Littlewood prime-tuple conjecture.
[16] viXra:1003.0179 [pdf] replaced on 16 Jul 2010
Authors: Jongsoo Park
Comments: 76 pages
Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. [14] Four parallel proofs of Szemer'edi's theorem have been achieved; one by direct combinatorics, one by ergodic theory, one by hypergraph theory, and one by Fourier analysis and additive combinatorics. Even with so many proofs, Professor T. Tao points out that with this problem, there remains a sense that our understanding of this result is incomplete; for instance, none of the approaches were powerful enough to detect progressions in the primes, mainly due to the sparsity of the prime sequence. [22] Oliver Lonsdale Atkin introduced a prime sieve using irreducible binary quadratic forms and modular arithmetic; the algorithm enumerates representations of integers by certain binary quadratic forms. A way that uses modular arithmetic is widely known: 6n+δ, 12n+δ, 30n+δ, 60n+δ.[31] In this paper, we assert that the composite number of the 12n+1, 5, 7, 11 series as selected by a Modular Arithmetic and Multiplication Table are not random but consist of very structural and regular arithmetic progression groups.
[15] viXra:1003.0179 [pdf] replaced on 2 Apr 2010
Authors: Jongsoo Park
Comments: 30 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.
[14] viXra:1003.0179 [pdf] replaced on 16 Mar 2010
Authors: Jongsoo Park
Comments: 30 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.
[13] viXra:1003.0089 [pdf] replaced on 12 May 2010
Authors: Stein E. Johansen
Comments: 41 pages, Submitted to Journal of Calcutta Mathematical Society, Nov 18, 2009.
We present a certain geometrical interpretation of the natural numbers, where these numbers appear as joint products of 5- and 3-multiples located at specified positions in a revolving chamber. Numbers without factors 2, 3 or 5 appear at eight such positions, and any prime number larger than 7 manifests at one of these eight positions after a specified amount of rotations of the chamber. Our approach determines the sets of rotations constituting primes at the respective eight positions, as the complements of the sets of rotations constituting non-primes at the respective eight positions. These sets of rotations constituting non-primes are exhibited from a basic 8x8-matrix of the mutual products originating from the eight prime numbers located at the eight positions in the original chamber. This 8x8-matrix is proven to generate all non-primes located at the eight positions in strict rotation regularities of the chamber. These regularities are expressed in relation to the multiple 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.
[12] viXra:1003.0089 [pdf] replaced on 11 Mar 2010
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.
[11] viXra:1003.0004 [pdf] replaced on 8 Mar 2010
Authors: Young-Mook Kang
Comments: 6 pages, Submitted to annals of mathematics
A study of growth of M(x) as x → ∞ is one of the most useful approach to the Riemann hypophotesis(RH). It is very known that the RH is equivalent to which M(x) = O(x1/2+ε) for ε > 0. Also Littlewood proved that "the RH is equivalent to the statement that limx → ∞ M(x)x-1/2-ε = 0, for every ε > 0".[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.
[10] viXra:1003.0004 [pdf] replaced on 5 Mar 2010
Authors: Young-Mook Kang
Comments: 5 pages, Submitted to annals of mathematics
A study of growth of M(x) as x → ∞ is one of the most useful approach to the Riemann hypophotesis(RH). It is very known that the RH is equivalent to which M(x) = O(x1/2+ε) for ε > 0. Also Littlewood proved that "the RH is equivalent to the statement that limx → ∞ M(x)x-1/2-ε = 0, for every ε > 0".[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.
[9] viXra:1001.0042 [pdf] replaced on 28 Jun 2010
Authors: Jose Javier Garcia Moreta
Comments: 18 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)
[8] viXra:1001.0038 [pdf] replaced on 7 Mar 2010
Authors: Jose Javier Garcia Moreta
Comments: 8 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.
[7] viXra:1001.0038 [pdf] replaced on 8 Feb 2010
Authors: Jose Javier Garcia Moreta
Comments: 7 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.
[6] viXra:0912.0043 [pdf] replaced on 21 Dec 2009
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.
[5] viXra:0911.0002 [pdf] replaced on 22 Nov 2009
Authors: Kazuya Kawai
Comments: 2 pages
The mersenne prime number exists in infinity.
[4] viXra:0911.0002 [pdf] replaced on 12 Nov 2009
Authors: Kazuya Kawai
Comments: 2 pages
The mersenne prime number exists in infinity.
[3] viXra:0911.0002 [pdf] replaced on 5 Nov 2009
Authors: Kazuya Kawai
Comments: 2 pages
The mersenne prime number exists in infinity.
[2] viXra:0908.0091 [pdf] replaced on 25 Aug 2009
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. This also solves the problem of finding two polynomials of degree n that fully factorise into linear factors with integer coefficients such that the difference is one.
[1] viXra:0812.0004 [pdf] replaced on 29 Dec 2008
Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org
see paper