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Arithmetic properties of Apéry-like numbers
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 2 / February 2018
- Published online by Cambridge University Press:
- 20 October 2017, pp. 249-274
- Print publication:
- February 2018
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- Article
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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 
$p$.
