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On divisors of Lucas and Lehmer numbers

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.

preprint2010arXivOpen access
C. L. Stewart
math.NT
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C. L. Stewart

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