The new Fermat-type factorization algorithm

Abstract

Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the Fermat's factorization algorithm [1 p.6,3 p.1470], implementation technique which I call the Fermat's factorization method (like the title, translated into English, of the reference document [2] published in Italian) and which is, among the implementation techniques [1 pp.6-8,2,3 pp.1470-1471] of the Fermat's factorization algorithm, the one with which a smaller iterations number occurs to identify the factors, trivial or non-trivial, of n (except for the circumstance in which two factors, trivial or non-trivial, of n are so close to each other that they are identified at the 1st iteration with each of the implementation techniques of the Fermat's factorization algorithm). In fact, through the way in which the Euler's function [4] is applied to the Fermat's factorization method, we arrive at this new factorization algorithm with which we obtain the certain reduction in the iterations number (except for the cases in which two factors of n are so close to each other that they are identified at the 1st iteration with the Fermat's factorization method) compared to the iterations number that occurs with the Fermat's factorization method. Furthermore, in this article I represent the hypotheses field according to which it will eventually be possible to further reduce the iterations number. Finally and always in relation to this new factorization algorithm, in this article I represent in detail the limit iterations number, which is smaller than the iterations number that occurs to reach the condition x - y = 1 which characterizes the pair of trivial factors of n, beyond which it is no longer possible for pairs of non-trivial factors of n to occur.