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We give a family of congruences for the binomial coefficients ${kp-1\choosep-1}$ in terms of multiple harmonic sums, a generalization of the harmonicnumbers. Each congruence in this family (which depends on an additionalparameter $n$) involves a linear combination of $n$ multiple harmonic sums, andholds $\mod{p^{2n+3}}$. The coefficients in these congruences are integersdepending on $n$ and $k$, but independent of $p$. More generally, we constructa family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose memberscontain a variable number of terms, and show that in this family there is aunique "optimized" congruence involving the fewest terms. The special case$k=2$ and $n=0$ recovers Wolstenholme's theorem ${2p-1\choose p-1}\equiv1\mod{p^3}$, valid for all primes $p\geq 5$. We also characterize those triples$(n, k, p)$ for which the optimized congruence holds modulo an extra power of$p$: they are precisely those with either $p$ dividing the numerator of theBernoulli number $B_{p-2n-k}$, or $k \equiv 0, 1 \mod p$.

Multiple harmonic sums and Wolstenholme's theorem