A prime number p is said to be a Wolstenholme prime if it satisfies the
congruence (pβ12pβ1β)β‘1(modp4). For such a prime
p, we establish the expression for (pβ12pβ1β)(modp8)
given in terms of the sums Riβ:=βk=1pβ1β1/ki (i=1,2,3,4,5,6).
Further, the expression in this congruence is reduced in terms of the sums
Riβ (i=1,3,4,5). Using this congruence, we prove that for any Wolstenholme
prime, (pβ12pβ1β)β‘1β2pk=1βpβ1βk1ββ2p2k=1βpβ1βk21β(modp7). 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.Comment: pages 1