On the $\mod p^7$ determination of ${2p-1\choose p-1}$

Abstract

In this paper we prove that for any prime $p\ge 11$, $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i\lt j\le p-1}\frac{1}{ij}\pmod{p^7} $$ holds. This is a generalization of the famous Wolstenholme's theorem which asserts that ${2p-1\choose p-1} \equiv 1 \ \pmod{\,p^3}$ for all primes $p\ge 5$. Our proof is elementary, and it does not use a standard technique involving the classic formula for power sums in terms of the Bernoulli numbers. Notice that the above congruence reduced modulo $p^6$, $p^5$ and $p^4$ yields related congruences obtained by Tauraso, Zhao and Glaisher, respectively.