A q-analog of Ljunggren's binomial congruence

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $$\binom{a p}{b p} \equiv \binom{a}{b}$$ modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.Comment: 6 pages, to be published in the proceedings of FPSAC 201

Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$1+\frac 12 +\frac 13+...+\frac{1}{p-1}$$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2}$$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type congruences (1862-2012) including more than 70 related results and 106 references. This is in fact version 2 of the paper extended with congruences (12) and (13

Multiple harmonic sums and Wolstenholme's theorem

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mod{p^{2n+3}}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose members contain a variable number of terms, and show that in this family there is a unique "optimized" congruence involving the fewest terms. The special case $k=2$ and $n=0$ recovers Wolstenholme's theorem ${2p-1\choose p-1}\equiv 1\mod{p^3}$, 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 the Bernoulli number $B_{p-2n-k}$, or $k \equiv 0, 1 \mod p$.Comment: 22 page