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

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 < 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.