On Wolstenholme's theorem and its converse

Abstract

For any positive integer n, let w n = ( 2 n − 1 n − 1 ) = 1 2 ( 2 n n ) . Wolstenholme proved that if p is a prime ⩾5, then w p ≡ 1 ( mod p 3 ) . The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers w n , and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers.