On divisors of Lucas and Lehmer numbers

Abstract

Let un be the nth term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of un which is of the form n exp(log n/104 log log n). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdo{double acute}s from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912. © 2013 Institut Mittag-Leffler.