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

TL;DR

This paper extends Wolstenholme's theorem by establishing a congruence for the binomial coefficient ${2p-1race p-1}$ modulo $p^7$, using elementary methods without Bernoulli numbers.

Contribution

It provides a new elementary proof of a higher-order congruence for ${2p-1race p-1}$ modulo $p^7$, generalizing previous results.

Findings

Proves a congruence modulo p^7 for ${2p-1race p-1}$.

Derives related congruences modulo p^6, p^5, and p^4.

Offers an elementary proof avoiding Bernoulli numbers.

Abstract

In this paper we prove that for any prime $p\ge 11$ holds $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i<j\le p-1}\frac{1}{ij}\pmod{p^7}. $$ This is a generalization of the famous Wolstenholme's theorem which asserts that ${2p-1\choose p-1} \equiv 1 \,\,(\bmod\,\,p^3)$ 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.