A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function

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Abstract

Fernando Rodriguez-Villegas has been studying hypergeometric families of Calabi–Yau manifolds, and from his investigations he has found (numerically) many possible supercongruences. For example, he conjectures for every odd prime p thatn=0p−12nn216−n−4p(modp2).Here, we use the theory of Gaussian hypergeometric series, the properties of the p-adic Γ-function, and a strange combinatorial identity to prove this conjecture.

MSC

11F85
11L10

Keywords

Supercongruences

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