This paper presents particle swarm optimization (PSO) method to find the prime factors of a composite number. Integer factorization is a well known NP hard problem and security of many cryptosystem is based on difficulty of integer factorization. A particle swarm optimization algorithm for integer factorization has been devised and tested on different 100 numbers. It has been found that the PSO...

In a previous paper, we derived a recursive formula determining the weight distributions of the [n = (qm − 1)/(q − 1), n−m, 3] Hamming code H(m,q), when (m, q−1) = 1. Here q is a prime power. We note here that the formula actually holds for any positive integer m and any prime power q, without the restriction (m,q − 1) = 1.

Let p be a prime and p1, . . . , pr be distinct prime divisors of p− 1. We prove that the smallest positive integer n which is a simultaneous p1, . . . , pr-power nonresidue modulo p satisfies n < p 1/4−cr+o(1) (p → ∞) for some positive cr satisfying cr = e −(1+o(1))r (r → ∞). Mathematical Subject Classification: 11A15, 11A07, 11N29

A polynomial P (x) with integer coefficients is said to be transitive modulo m, if for every x, y ∈ Z there exists k ≥ 0 such that P (x) = y (mod m). In this paper, we construct new examples of transitive polynomials modulo prime powers and partially describe cubic and quartic transitive polynomials. We also study the orbit structure of affine maps modulo prime powers.

For a given prime p, what is the smallest integer n such that there exists a group of order p in which the set of commutators does not form a subgroup? In this paper we show that n = 6 for any odd prime and n = 7 for p = 2.

We study a relation between factorials and their additive analog, the triangular numbers. We show that there is a positive integer k such that n! = 2kT where T is a product of triangular numbers. We discuss the primality of T±1 and the primality of |T − p| where p is either the smallest prime greater than T or the greatest prime less than T .

Let /(x) be a polynomial with integer coefficients, and let D(/)-gx.d.{/.(*):*eZ}. It was conjectured by Bouniakowsky in 1857 that if f(x) is nonconstant and irreducible over Z, theii \f(x)\/D(f) is prime for infinitely many integers x. It is shown that there exist irreducible polynomials f(x) with D(f) = 1 such that the smallest integer x for which \f(x)\ is prime is large as a function of the...

Given a positive integer n, let P(n) denote the largest prime factor of nand S(n) denote the smallest integer m such that n divides m! This paper extends earlier work [1] on the average value of the Smarandache function S(n) and is based on a recent asymptotic result [2]: (W J I{n ~ N:P(n) < S(n)}j = 01 • I ~ln(N)J for any positive integer j due to Ford. We first prove: Theorem 1.

Let N(a, m) be the least integer n (if exists) such that φ(n) ≡ a (mod m). Friedlander and Shparlinski proved that for any ε > 0 there exists A = A(ε) > 0 such that for any positive integer m which has no prime divisors p < (log m) and any integer a with gcd(a,m) = 1, we have the bound N(a,m) ¿ m. In the present paper we improve this bound to N(a,m) ¿ m.

Let G be a transitive permutation group on a set ? and let m be a positive integer. If no element of G moves any subset of ? by more than m points, then |? | [2mp I (p-1)] wherep is the least odd primedividing |G |. When the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, I i n....