An algorithm is presented which, given a positive integer n, will either factor n or prove it to be prime. The algorithm takes O(«) steps.

Let p be a prime number. Fermat's little theorem [1] states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between 1 and p-1. A primitive root [1] of p is a number r such that any integer a between 1 and p-1 can be expressed by a=r^k mod p, with k a nonnegative integer smaller that p-1. If p is an odd prime number then r is a primitive root of p if and only if r^((p-...

We announce some new results about systems of residue sets. A residue set R C Z is an arithmetic progression R = {a,a±n,a± 2n,...}. The positive integer n is referred to as the modulus of R. Following Znam [21] we denote this set by a(n). We need several number-theoretic functions. p(ra)-the least prime divisor of a natural number ra, P(m)-the greatest prime divisor of ra, A(m)-the greatest div...

Prime numbers are extremely useful, and are an essential input to many algorithms in large part due to the algebraic structure of arithmetic modulo a prime. In everyday life, perhaps the most frequent use for prime numbers is in RSA encryption, which requires quite large primes (typically≥ 128-bits long). Fortunately, there are lots of primes—for large n, the probability that a random integer l...

these are the positive integers that each have only two integer divisors (namely 1 and the number itself). By convention we do not take the number 1 to be prime. An immediate natural question, to which this notion leads, is already rather difficult: is the set of all the primes finite? In other words, can the above sequence of primes be continued indefinitely? The answer to this question was di...

Given an integer γ≥2 and odd prime power q we show that for every large genus g there exists a non-singular curve C defined over

This is a pioneering work, introducing novel class of optimization objective functions over subsets primeonly integer points. We show rich variety Prime Optimization and mixed problems.

Let q be a prime power and define (n) =1+q+q 2 +⋯+q n-1 , for nonnegative integer n. B denote the set of all subspaces

in this paper we classify fuzzy subgroups of a rank-3 abelian group $g = mathbb{z}_{p^n} + mathbb{z}_p + mathbb{z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. we present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...

Given an integer n ≥ 2, let p m (n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of p m (n) for n ≤ x.