On divisors of Lucas and Lehmer numbers

Cameron L. Stewart

doi:10.1007/s11511-013-0105-y

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Home > Journals > Acta Math. > Volume 211 > Issue 2 > Article

2013 On divisors of Lucas and Lehmer numbers

Cameron L. Stewart

Author Affiliations +

Cameron L. Stewart¹
¹Department of Pure Mathematics, University of Waterloo

Acta Math. 211(2): 291-314 (2013). DOI: 10.1007/s11511-013-0105-y

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Abstract

Let u_n 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 u_n 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 Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

Funding Statement

Research supported in part by the Canada Research Chairs Program and by Grant A3528 from the Natural Sciences and Engineering Research Council of Canada.