# Wolstenholme prime

Named after Joseph Wolstenholme 1995[1] McIntosh, R. J. 2 Infinite Irregular primes 16843, 2124679 2124679

In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century.

Interest in these primes first arose due to their connection with Fermat's last theorem, another theorem with significant importance in mathematics. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two.

The only two known Wolstenholme primes are 16843 and 2124679 (sequence A088164 in the OEIS). There are no other Wolstenholme primes less than 109.[2]

## Definition

 Unsolved problem in mathematics:Are there any Wolstenholme primes other than 16843 and 2124679?(more unsolved problems in mathematics)

Wolstenholme prime can be defined in a number of equivalent ways.

### Definition via binomial coefficients

A Wolstenholme prime is a prime number p > 7 that satisfies the congruence

${\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}$

where the expression in left-hand side denotes a binomial coefficient.[3] Compare this with Wolstenholme's theorem, which states that for every prime p > 3 the following congruence holds:

${\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}$

### Definition via Bernoulli numbers

A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3.[4][5][6] The Wolstenholme primes therefore form a subset of the irregular primes.

### Definition via irregular pairs

A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.[7][8]

### Definition via harmonic numbers

A Wolstenholme prime is a prime p such that[9]

${\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}$

i.e. the numerator of the harmonic number ${\displaystyle H_{p-1}}$ expressed in lowest terms is divisible by p3.