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ON THE LARGEST PRIME FACTOR OF THE MERSENNE NUMBERS

Part of: Multiplicative number theory Sequences and sets

Published online by Cambridge University Press:  17 April 2009

KEVIN FORD ,
FLORIAN LUCA  and
IGOR E. SHPARLINSKI
KEVIN FORD
Affiliation:
Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana, Champaign, IL 61801, USA (email: [email protected])
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México (email: [email protected])
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series is convergent for each constant α<1/2, which gives a more precise form of a result of C. L. Stewart [‘On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers’, Proc. London Math. Soc.35(3) (1977), 425–447].

Keywords

primesMersenne numbersapplications of sieve methods

MSC classification

Secondary: 11B83: Special sequences and polynomials 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 455 - 463
Copyright
Copyright © Australian Mathematical Society 2009

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