Congruences for Wolstenholme primes

Abstract

A prime number $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \,\,(\bmod{\,\,p^4})$. For such a prime $p$, we establish the expression for ${2p-1\choose p-1}\,\,(\bmod{\,\,p^8})$ given in terms of the sums $R_i:=\sum_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime, $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} -2p^2\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^7}. $$ Moreover, using a recent result of the author \cite{Me}, we prove that the above congruence implies that a prime $p$ necessarily must be a Wolstenholme prime. Applying a technique of Helou and Terjanian \cite{HT}, the above congruence is given as the expression involving the Bernoulli numbers.