Arithmetic properties of Apéry-like numbers. (20th October 2017) - Delaygue, É.
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$p$ .