A Simple Algorithm for Prime Factorization and Primality Testing

Abstract

We propose a new simple and faster algorithm to factor numbers based on the nature of the prime numbers contained in such composite numbers. It is well known that every composite number has a unique representation as a product of prime numbers. In this study, we focus mainly on composite numbers that contain a product of prime numbers that are greater than or equal to 5 which are of the form $6k+1$ or $6k+5$ . Therefore, we use the condition that every prime or composite $P$ of primes greater than or equal to 5 satisfies $P^2\equiv 1\hspace{1em}\left(\mathrm{mod}24\right)$ . This algorithm is very fast especially when the difference in the prime components of a composite number (prime gap) is not so large. When the difference between the factors (prime gap) is not so large, it often requires just a single iteration to obtain the factors.