On the mod $p^7$ determination of $\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)$

Abstract

In this paper we prove that for any prime $p\ge 11$ holds $$\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)\equiv 1-2p\sum _{k=1}^{p-1}\frac{1}{k}+4p^2\sum _{1\le i<j\le p-1}\frac{1}{ij}(\mathrm{mod}{p}^7).$$ This is a generalization of the famous Wolstenholme's theorem which asserts that $\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)\equiv 1(\mathrm{mod}{p}^3)$ for all primes $p\ge 5$. Our proof is elementary and it does not use a standard technique involving the classic formula for the 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 R. Tauraso, J. Zhao and J.W.L. Glaisher, respectively.