Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T13:39:49.903Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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