Arithmetic properties of Apéry-like numbers. (20th October 2017)

• Record Type: Journal Article
• Authors: Delaygue, É.
• Is Part Of: Compositio mathematica. Volume 154:Number 2(2018)
• Page Start: 249
• Page End: 274
• Publication Date: 2017-10-20
• Subjects: 11B50 (primary), -- 11B65, -- 05A10 (secondary); Apéry numbers, -- constant terms of powers of Laurent polynomials, -- p-Lucas property, -- congruences; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007552
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We provide lower bounds for$p$ -adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the$p$ -adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the$p$ -Lucas property for almost all primes$p$ .