Wolstenholme prime
| Named after | Joseph Wolstenholme |
|---|---|
| Publication year | 1995 |
| Author of publication | McIntosh, R. J. |
| Number of known terms | 2 |
| Conjectured number of terms | Infinite |
| Subsequence of | Irregular primes |
| First terms | 16843, 2124679 |
| Largest known term | 2124679 |
| OEIS index | A088164 |
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 7. 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 in the OEIS). There are no other Wolstenholme primes less than 109.
Wolstenholme prime can be defined in a number of equivalent ways.
A Wolstenholme prime is a prime number p > 7 that satisfies the congruence
where the expression in left-hand side denotes a binomial coefficient. Compare this with Wolstenholme's theorem, which states that for every prime p > 3 the following congruence holds:
A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3. The Wolstenholme primes therefore form a subset of the irregular primes.
A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.
A Wolstenholme prime is a prime p such that
i.e. the numerator of the harmonic number expressed in lowest terms is divisible by p3.
...
Wikipedia