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. 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 1011.
Definition
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
( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ) , {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}
where the expression in left-hand side denotes a binomial coefficient. In comparison, Wolstenholme's theorem states that for every prime p > 3 the following congruence holds:
( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) . {\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, or equivalently,
B p − 3 ≡ 0 ( mod p ) {\displaystyle B_{p-3}\equiv 0{\pmod {p}}}
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.
Definition via harmonic numbers A Wolstenholme prime is a prime p such that
H p − 1 ≡ 0 ( mod p 3 ) , {\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}
i.e. the numerator of the harmonic number H p − 1 {\displaystyle H_{p-1}} expressed in lowest terms is divisible by p3.
Properties A prime p > 7 is a Wolstenholme prime if and only if
∑ k = ⌊ p / 6 ⌋ + 1 ⌊ p / 4 ⌋ 1 k 3 ≡ 0 ( mod p ) {\displaystyle \sum _{k=\lfloor p/6\rfloor +1}^{\lfloor p/4\rfloor }{\frac {1}{k^{3}}}\equiv 0{\pmod {p}}}
Search and current status The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2022. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time. The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993. Up to 1.2×107, no further Wolstenholme primes were found. This was later extended to 2×108 by McIntosh in 1995 and Trevisan and Weber were able to reach 2.5×108. The latest result as of 2022 is that there are only those two Wolstenholme primes up to 1011.
Expected number of Wolstenholme primes It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes less than x {\displaystyle x} is about log log x {\displaystyle \log \log x} . For each prime p ≥ 5 {\displaystyle p\geq 5} , the Wolstenholme quotient is defined as
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