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On divisors of Lucas and Lehmer numbers

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Acta Mathematica

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.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Cameron L. Stewart

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Correspondence to Cameron L. Stewart.

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

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Stewart, C.L. On divisors of Lucas and Lehmer numbers. Acta Math 211, 291–314 (2013). https://doi.org/10.1007/s11511-013-0105-y

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  • DOI: https://doi.org/10.1007/s11511-013-0105-y

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