A detrministic factorization and primality testing algorithm for integers of the form Z mod 6 = -1

Abstract

Prime numbers are known to be in one of two series; P mod 6 = ±1. In this paper, we introduce the concept of Integer Absolute Position in prime series, and we use the concept to develop a structure for composite integer numbers in the prime series P mod 6 = -1. We use the developed structure to state theorems and to develop a deterministic algorithm that can test simultaneously for primality and prime factors of integers of the form Z mod 6 = -1. The developed algorithm is compared with some of the well known factorization algorithms. The results show that the developed algorithm performs well when the two factors are close to each other. Although the current version of the algorithm is of complexity ((N/62) 1/2 /2), but the facts that, the algorithm has a parallel characteristics and its performance is dependent on a matrix search algorithm, makes the algorithm competent for achieving better performance. © 2011 Springer-Verlag.