Wolstenholme prime

Wolstenholme prime
Named after	Joseph Wolstenholme
Publication year	1995[1]
Author of publication	McIntosh, R. J.
No. of known terms	2
Conjectured no. of terms	Infinite
Subsequence of	Irregular primes
First terms	16843, 2124679
Largest known term	2124679
OEIS index	A088164; Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4);

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 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

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

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

{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 B_p−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]

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

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

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 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.[10] 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.[11] Up to 1.2×10⁷, no further Wolstenholme primes were found.[12] This was later extended to 2×10⁸ by McIntosh in 1995 [5] and Trevisan & Weber were able to reach 2.5×10⁸.[13] The latest result as of 2007 is that there are only those two Wolstenholme primes up to 10⁹.[14]

Expected number of Wolstenholme primes

It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ x is about ln ln x, where ln denotes the natural logarithm. For each prime p ≥ 5, the Wolstenholme quotient is defined as

W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.

Clearly, p is a Wolstenholme prime if and only if W_p ≡ 0 (mod p). Empirically one may assume that the remainders of W_p modulo p are uniformly distributed in the set {0, 1, ..., p–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/p.[5]

Notes

Wolstenholme primes were first described by McIntosh in McIntosh 1995, p. 385
Weisstein, Eric W. "Wolstenholme prime". MathWorld.
Cook, J. D. "Binomial coefficients". Retrieved 21 December 2010.
Clarke & Jones 2004, p. 553.
McIntosh 1995, p. 387.
Zhao 2008, p. 25.
Johnson 1975, p. 114.
Buhler et al. 1993, p. 152.
Zhao 2007, p. 18.
Selfridge and Pollack published the first Wolstenholme prime in Selfridge & Pollack 1964, p. 97 (see McIntosh & Roettger 2007, p. 2092).
Ribenboim 2004, p. 23.
Zhao 2007, p. 25.
Trevisan & Weber 2001, p. 283–284.
McIntosh & Roettger 2007, p. 2092.

References

Selfridge, J. L.; Pollack, B. W. (1964), "Fermat's last theorem is true for any exponent up to 25,000", Notices of the American Mathematical Society, 11: 97
Johnson, W. (1975), "Irregular Primes and Cyclotomic Invariants" (PDF), Mathematics of Computation, 29 (129): 113–120, doi:10.2307/2005468, JSTOR 2005468 Archived 2010-12-20 at WebCite
Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T. (1993), "Irregular Primes and Cyclotomic Invariants to Four Million" (PDF), Mathematics of Computation, 61 (203): 151–153, Bibcode:1993MaCom..61..151B, doi:10.2307/2152942, JSTOR 2152942 Archived 2010-11-12 at WebCite
McIntosh, R. J. (1995), "On the converse of Wolstenholme's Theorem" (PDF), Acta Arithmetica, 71 (4): 381–389, doi:10.4064/aa-71-4-381-389 Archived 2010-11-08 at WebCite
Trevisan, V.; Weber, K. E. (2001), "Testing the Converse of Wolstenholme's Theorem" (PDF), Matemática Contemporânea, 21: 275–286 Archived 2010-12-10 at WebCite
Ribenboim, P. (2004), "Chapter 2. How to Recognize Whether a Natural Number is a Prime", The Little Book of Bigger Primes, New York: Springer-Verlag New York, Inc., ISBN 978-0-387-20169-6 Archived 2010-11-24 at WebCite
Clarke, F.; Jones, C. (2004), "A Congruence for Factorials" (PDF), Bulletin of the London Mathematical Society, 36 (4): 553–558, doi:10.1112/S0024609304003194 Archived 2011-01-02 at WebCite
McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, doi:10.1090/S0025-5718-07-01955-2 Archived 2010-12-10 at WebCite
Zhao, J. (2007), "Bernoulli numbers, Wolstenholme's theorem, and p⁵ variations of Lucas' theorem" (PDF), Journal of Number Theory, 123: 18–26, doi:10.1016/j.jnt.2006.05.005, S2CID 937685Archived 2010-11-12 at WebCite
Zhao, J. (2008), "Wolstenholme Type Theorem for Multiple Harmonic Sums" (PDF), International Journal of Number Theory, 4 (1): 73–106, doi:10.1142/s1793042108001146 Archived 2010-11-27 at WebCite

External links

Caldwell, Chris K. Wolstenholme prime from The Prime Glossary
McIntosh, R. J. Wolstenholme Search Status as of March 2004 e-mail to Paul Zimmermann
Bruck, R. Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients
Conrad, K. The p-adic Growth of Harmonic Sums interesting observation involving the two Wolstenholme primes

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[1] Wolstenholme primes were first described by McIntosh in McIntosh 1995, p. 385

[2] Weisstein, Eric W. "Wolstenholme prime". MathWorld.

[3] Cook, J. D. "Binomial coefficients". Retrieved 21 December 2010.

[FOOTNOTEClarkeJones2004553-4] Clarke & Jones 2004, p. 553.

[FOOTNOTEMcIntosh1995387-5] McIntosh 1995, p. 387.

[FOOTNOTEZhao200825-6] Zhao 2008, p. 25.

[FOOTNOTEJohnson1975114-7] Johnson 1975, p. 114.

[FOOTNOTEBuhlerCrandallErnvallMetsänkylä1993152-8] Buhler et al. 1993, p. 152.

[FOOTNOTEZhao200718-9] Zhao 2007, p. 18.

[10] Selfridge and Pollack published the first Wolstenholme prime in Selfridge & Pollack 1964, p. 97 (see McIntosh & Roettger 2007, p. 2092).

[FOOTNOTERibenboim200423-11] Ribenboim 2004, p. 23.

[FOOTNOTEZhao200725-12] Zhao 2007, p. 25.

[FOOTNOTETrevisanWeber2001283–284-13] Trevisan & Weber 2001, p. 283–284.

[FOOTNOTEMcIntoshRoettger20072092-14] McIntosh & Roettger 2007, p. 2092.

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Weakly Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
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Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers