Arithmetic properties of Apéry-like numbers

Eric Delaygue (1)

1 CTN - Combinatoire, théorie des nombres

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, 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.

Mots clés: Apéry numbers; Constant terms of powers of Laurent polynomials; p-Lucas property; Congruences

Domaines: Théorie des nombres [math.NT]

Eric Delaygue. Arithmetic properties of Apéry-like numbers. Compositio Mathematica, 2018, 154 (2). ⟨hal-00873552v2⟩