On divisors of Lucas and Lehmer numbers

TL;DR

This paper establishes a lower bound for the greatest prime factor of Lucas and Lehmer sequence terms, resolving longstanding questions and improving historical results on prime factors of such sequences.

Contribution

It provides the first general lower bound for prime factors of Lucas and Lehmer numbers, answering questions posed by Schinzel and Erdos.

Findings

Proved a lower bound of the form n*exp(log n / 104 log log n) for prime factors.

Resolved a question of Schinzel from 1962.

Improved upon results of Bang and Carmichael from the late 19th and early 20th centuries.

Abstract

Let u(n) be the n-th 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 u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.