An elementary proof of a Rodriguez-Villegas supercongruence

Abstract

We give a short proof of the following known congruence: for every odd prime $p$ $$\sum _{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^{2}16^{-k}\equiv (-1{)}^{\frac{p-1}{2}}(\mathrm{mod}{p}^2).$$ Moreover, we provide some new results connected with the above congruence.

Cited by in corpus (8)

• Some q-analogues of supercongruences of Rodriguez-Villegas
• New congruences involving products of two binomial coefficients
• Super congruences involving Bernoulli and Euler polynomials
• Congruences involving binomial coefficients and Lucas sequences
• Congruences for truncated hypergeometric series ${}_2{F}_1$
• Supercongruences involving $p$-adic Gamma functions
• Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas
• Factors of some truncated basic hypergeometric series