Arithmetic properties of Apéry numbers

Florian Luca; Igor E. Shparlinski

doi:10.1112/jlms/jdn031

Arithmetic properties of Apéry numbers

Florian Luca^*, Igor E. Shparlinski

^*Corresponding author for this work

Department of Computing

Research output: Contribution to journal › Article

2 Citations (Scopus)

Abstract

Let (An)n≥1 be the sequence of Apéry numbers with a general term given by. In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)^1/2 hold for a set of positive integers n of asymptotic density 1. Here, ω(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 and certain Lucas-type congruences.

Original language	English
Pages (from-to)	545-562
Number of pages	18
Journal	Journal of the London Mathematical Society
Volume	78
Issue number	3
DOIs	https://doi.org/10.1112/jlms/jdn031
Publication status	Published - Dec 2008

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title = "Arithmetic properties of Ap{\'e}ry numbers",

abstract = "Let (An)n≥1 be the sequence of Ap{\'e}ry numbers with a general term given by. In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, ω(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 and certain Lucas-type congruences.",

author = "Florian Luca and Shparlinski, {Igor E.}",

year = "2008",

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volume = "78",

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AU - Shparlinski, Igor E.

PY - 2008/12

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N2 - Let (An)n≥1 be the sequence of Apéry numbers with a general term given by. In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, ω(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 and certain Lucas-type congruences.

AB - Let (An)n≥1 be the sequence of Apéry numbers with a general term given by. In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, ω(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 and certain Lucas-type congruences.

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DO - 10.1112/jlms/jdn031

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VL - 78

SP - 545

EP - 562

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JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

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