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[39] 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.
[38] viXra:1003.0102 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
On Carmichaël's conjecture
[37] viXra:1003.0095 [pdf] submitted on 6 Mar 2010
Authors: Mihàly 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].
[36] 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".
[35] 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.
[34] 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)
[33] 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.
[32] 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.
[31] 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.
[30] 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.
[29] 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.
[28] 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.
[27] 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.
[26] 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.
[25] 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.
[24] viXra:1002.0026 [pdf] submitted on 15 Feb 2010
Authors: Kazuya Kawai
Comments: 1 pages
The method is efficiently good at factorization on prime numbers from existing methods.
[23] 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.
[22] viXra:1001.0047 [pdf] submitted on 29 Jan 2010
Authors: Jose Javier Garcia
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)
[21] viXra:1001.0042 [pdf] submitted on 27 Jan 2010
Authors: Jose Javier Garcia
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)
[20] viXra:1001.0039 [pdf] submitted on 26 Jan 2010
Authors: Jose Javier Garcia
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.
[19] viXra:1001.0038 [pdf] submitted on 26 Jan 2010
Authors: Jose Javier Garcia
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.
[18] 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.
[17] 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
[16] 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.
[15] viXra:0911.0002 [pdf] submitted on 2 Nov 2009
Authors: Kazuya Kawai
Comments: 2 pages
The mersenne prime number exists in infinity.
[14] viXra:0910.0019 [pdf] submitted on 13 Oct 2009
Authors: Hideyuki Ohtsuka
Comments: 3 Pages
In this paper, we show relations among sums of powers of Fibonacci numbers.
[13] viXra:0910.0012 [pdf] submitted on 9 Oct 2009
Authors: Hideyuki Ohtsuka, Shigeru Nakamura
Comments: 3 Pages, This article will appear in Application of Fibonacci Numbers, Volume 12.
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. Submitted to INTEGERS: 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
[12] 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.
[11] 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.
[10] viXra:1002.0026 [pdf] replaced on 25 Feb 2010
Authors: Kazuya Kawai
Comments: 1 page
The method is efficiently good at factorization on prime numbers from existing methods.
[9] viXra:1002.0026 [pdf] replaced on 17 Feb 2010
Authors: Kazuya Kawai
Comments: 1 pages
The method is efficiently good at factorization on prime numbers from existing methods.
[8] viXra:1001.0038 [pdf] replaced on 7 Mar 2010
Authors: Jose Javier Garcia
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
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