Acta Mathematica

On divisors of Lucas and Lehmer numbers

Cameron L. Stewart

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

Note

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

Article information

Source
Acta Math., Volume 211, Number 2 (2013), 291-314.

Dates
Received: 2 February 2012
Revised: 16 November 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892739

Digital Object Identifier
doi:10.1007/s11511-013-0105-y

Mathematical Reviews number (MathSciNet)
MR3143892

Zentralblatt MATH identifier
1362.11070

Rights
2013 © Institut Mittag-Leffler

Citation

Stewart, Cameron L. On divisors of Lucas and Lehmer numbers. Acta Math. 211 (2013), no. 2, 291--314. doi:10.1007/s11511-013-0105-y. https://projecteuclid.org/euclid.acta/1485892739


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