The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

Abstract. Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p − 1 exceeding (p − 1) 1 2 in which the constants are effectively computable. As a result we prove that it is possible to calculate a value x0 such that for every x>x0 there is a p<xwith the greatest prime factor of p − 1 exceeding x 3 5. The novelty of our approach is the avoidance of any appeal to Siegel’s Theorem on primes in arithmetic progression. 1.