Three pairs of congruences concerning sums of central binomial coefficients

Abstract

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ${\sum }_{k=1}^{\lfloor \frac{2p}{3}\rfloor }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\equiv 0(\mathrm{mod}{p}^2)$ for any prime $p=1(\mathrm{mod}3)$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum _{k=1}^{\lfloor \frac{ap}{r}\rfloor }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)x^k(\mathrm{mod}{p}^2)$$ where $p$ is a prime such that $p\equiv 1(\mathrm{mod}r)$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions ${\Gamma }_p$ and a special class of computer-discovered hypergeometric identities.