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ON THE LARGEST PRIME FACTOR OF THE MERSENNE NUMBERS

Part of: Sequences and sets Multiplicative number theory

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: ford@math.uiuc.edu)
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México (email: fluca@matmor.unam.mx)
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor@ics.mq.edu.au)
*
✉For correspondence; e-mail: igor@ics.mq.edu.au
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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

References

[1] Birkhoff, G. D. and Vandiver, H. S., ‘On the integral divisors of a n−b n’, Ann. of Math. (2) 5 (1904), 173–180.CrossRefGoogle Scholar
[2] Canfield, E. R., Erdős, P. and Pomerance, C., ‘On a problem of Oppenheim concerning “factorisatio numerorum”’, J. Number Theory 17 (1983), 1–28.CrossRefGoogle Scholar
[3] Erdős, P., Kiss, P. and Pomerance, C., ‘On prime divisors of Mersenne numbers’, Acta Arith. 57 (1991), 267–281.CrossRefGoogle Scholar
[4] Erdős, P. and Shorey, T. N., ‘On the greatest prime factor of 2p−1 for a prime p and other expressions’, Acta Arith. 30 (1976), 257–265.CrossRefGoogle Scholar
[5] Halberstam, H. and Richert, H.-E., Sieve Methods (Academic Press, London, 1974).Google Scholar
[6] Hardy, G. H. and Ramanujan, S., ‘The normal number of prime factors of an integer’, Quart. J. Math. (Oxford) 48 (1917), 76–92.Google Scholar
[7] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th edn (Clarendon Press, Oxford, 1979).Google Scholar
[8] Hildebrand, A. and Tenenbaum, G., ‘Integers without large prime factors’, J. de Théorie des Nombres de Bordeaux 5 (1993), 411–484.Google Scholar
[9] Matveev, E. M., ‘An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II’, Izv. Ross. Akad. Nauk. Ser. Math. 64 (2000), 125–180; Engl. transl. Izv. Math. 64 (2000), 1217–1269.Google Scholar
[10] Murata, L. and Pomerance, C., On the Largest Prime Factor of a Mersenne Number, Number Theory CRM Proceedings Lecture Notes, 36 (American Mathematical Society, Providence, RI, 2004), pp. 209–218.Google Scholar
[11] Murty, R. and Wong, S., ‘The ABC conjecture and prime divisors of the Lucas and Lehmer sequences’, in: Number Theory for the Millennium, III, Urbana, IL, 2000 (A K Peters, Natick, MA, 2002), pp. 43–54.Google Scholar
[12] Pomerance, C., ‘On primitive divisors of Mersenne numbers’, Acta Arith. 46(4) (1986), 355–367.CrossRefGoogle Scholar
[13] Saias, E., ‘Entiers à diviseurs denses 1’, J. Number Theory 62 (1997), 163–191.CrossRefGoogle Scholar
[14] Schinzel, A., ‘On primitive prime factors of a n−b n’, Proc. Cambridge Philos. Soc. 58 (1962), 555–562.CrossRefGoogle Scholar
[15] Stewart, C. L., ‘The greatest prime factor of a n−b n’, Acta Arith. 26(4) (1974/75), 427–433.CrossRefGoogle Scholar
[16] Stewart, C. L., ‘On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers’, Proc. London Math. Soc. 35(3) (1977), 425–447.CrossRefGoogle Scholar
[17] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, Cambridge, 1995).Google Scholar