On the greatest prime factor of $p-1$ with effective constants

Abstract

Let p denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of p - I 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 x 0 such that for every x > x 0 there is a p < x with 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.