Some Observations on Primality Testing

Let N be an integer which is to be tested for primality. Previous methods of ascertaining the primality of N make use of factors of $N \pm 1$, ${N^2} \pm N + 1$, and ${N^2} + 1$ in order to increase the size of any possible prime divisor of N until it is impossible for N to be the product of two or more primes. These methods usually 2 work as long as $N < {K^2}$ , where K is $1/12$ of the product of the known prime power factors of $N \pm 1$, ${N^2} \pm N + 1$, and ${N^2} + 1$. In this paper a technique is described which, when used in conjunction with these methods, will often determine the pri mality of N when $N < l{K^3}$ and l is small.