Wolstenholme's theorem

For every prime p greater than 3, the binomial coefficient (2p−1 choose p−1) is congruent to 1 modulo p³. Equivalently, the sum of reciprocals 1 through p−1 is divisible by p² in the p-adic sense.

Connect

Binomial coefficient: A count of ways to choose a fixed-size subset from a finite set, also represented by a standard algebraic coefficient. The theorem concerns a particular binomial coefficient whose divisibility is unusually strong.
Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The modulus and the theorem's condition are both governed by a prime p.
Joseph Wolstenholme: A British mathematician and astronomer whose 1862 work established the congruence now bearing his name. His paper gave the result that became known as Wolstenholme's theorem.
Harmonic congruence: A congruence involving a finite sum of reciprocal powers of integers modulo a prime or its powers. The theorem is a landmark case for the sum of first reciprocal powers modulo p².
Wolstenholme prime conjecture: The conjectural claim that infinitely many primes satisfy the stronger Wolstenholme congruence modulo p⁴. Only two such primes are known, leaving their infinitude unresolved.
Congruence: An equality relation between integers that have the same remainder upon division by a fixed modulus. The theorem states equality modulo the prime cube, not equality of ordinary integers.
Modular arithmetic: Arithmetic in which integers are identified when they differ by a multiple of a fixed modulus. The theorem's conclusion is a modular equality modulo p³.
Wolstenholme primes: Primes p for which the binomial coefficient (2p−1 choose p−1) is congruent to 1 modulo p⁴. They strengthen the theorem's usual modulo-p³ conclusion by one power of p.
Bernoulli number: A rational number in a sequence defined by the generating function x/(e^x−1). Bernoulli numbers encode many higher reciprocal-power sums related to Wolstenholme-type congruences.
Prime-counting function: The function π(x) counts primes less than or equal to x. Counting exceptional primes asks how their number grows among all primes.