A Simple Algorithm for Prime Factorization and Primality Testing

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 
 
 6
 k
 +
 1
 
 or 
 
 6
 k
 +
 5
 
 . Therefore, we use the condition that every prime or composite 
 
 P
 
 of primes greater than or equal to 5 satisfies 
 
 
 
 P
 
 
 2
 
 
 ≡
 1
  
 
 
 mod
 24
 
 
 
 . 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.