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Journal of the London Mathematical Society Advance Access published online on July 8, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn031
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© 2008 London Mathematical Society

Arithmetic properties of Apéry numbers

Florian Luca

Instituto de Matemáticas
Universidad Nacional Autonoma de México
C.P. 58089, Morelia
Michoacán
México
fluca@matmor.unam.mx

Igor E. Shparlinski

Department of Computing Macquarie University
Sydney, NSW 2109
Australia

Let (An)n≥1 be the sequence of Apéry numbers and its general term is given by Formula . In this paper, we prove that both inequalities {omega}(An)>c0 log n log log n and P(An)>c0 (log nlog  log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, {omega}(m) is the number of distinct prime factors of m, P(m) is the largest prime factor of m and c0>0 is an absolute constant. The method applies to more general sequences satisfying both a linear recurrence of order 2 with polynomial coefficients as well as certain Lucas type congruences.

During the preparation of this paper, F.L. was also supported in part by grant SEP-CONACyT 46755 and I.S. was supported in part by ARC grant DP0556431.

Received August 8, 2007; revised March 11, 2008;
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