Wolstenholme prime#Expected number of Wolstenholme primes

{{Infobox integer sequence

| named_after = Joseph Wolstenholme

| publication_year = 1995Wolstenholme primes were first described by McIntosh in {{Harvnb|McIntosh|1995|p=385}}

| author = McIntosh, R. J.

| terms_number = 2

| con_number = Infinite

| parentsequence = Irregular primes

| first_terms = 16843, 2124679

| largest_known_term = 2124679

| OEIS = A088164

| OEIS_name = 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 {{OEIS|A088164 }}. There are no other Wolstenholme primes less than 10¹¹.{{MathWorld|urlname=WolstenholmePrime|title=Wolstenholme prime|mode=cs2|ref=none}}

Definition

{{unsolved|mathematics|Are there any Wolstenholme primes other than 16843 and 2124679?}}

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.{{citation| url = http://www.johndcook.com/binomial_coefficients.html | title = Binomial coefficients | first = J. D. | last = Cook | access-date = 21 December 2010}}

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.{{sfn|Clarke|Jones|2004|p=553}}{{sfn|McIntosh|1995|p=387}}{{sfn|Zhao|2008|p=25}} 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.{{sfn|Johnson|1975|p=114}}{{sfn|Buhler|Crandall|Ernvall|Metsänkylä|1993|p=152}}

=Definition via harmonic numbers=

A Wolstenholme prime is a prime p such that{{sfn|Zhao|2007|p=18}}

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

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 2022. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.Selfridge and Pollack published the first Wolstenholme prime in {{Harvnb|Selfridge|Pollack|1964|p=97}} (see {{Harvnb|McIntosh|Roettger|2007|p=2092}}). 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.{{sfn|Ribenboim|2004|p=23}} Up to 1.2{{e|7}}, no further Wolstenholme primes were found.{{sfn|Zhao|2007|p=25}} This was later extended to 2{{e|8}} by McIntosh in 1995{{sfn|McIntosh|1995|p=387}} and Trevisan & Weber were able to reach 2.5{{e|8}}.{{sfn|Trevisan|Weber|2001|p=283–284}} The latest result as of 2022 is that there are only those two Wolstenholme primes up to 10¹¹.{{Cite journal |last1=Booker |first1=Andrew R. |last2=Hathi |first2=Shehzad |last3=Mossinghoff |first3=Michael J. |last4=Trudgian |first4=Timothy S. | author4-link=Timothy Trudgian| date=2022-07-01 |title=Wolstenholme and Vandiver primes |url=https://link.springer.com/article/10.1007/s11139-021-00438-3 |journal=The Ramanujan Journal |language=en |volume=58 |issue=3 |pages=913–941 |doi=10.1007/s11139-021-00438-3 |issn=1572-9303|arxiv=2101.11157 }}

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.{{sfn|McIntosh|1995|p=387}}

Notes

References

{{Citation | last1=Buhler | first1=J. | last2=Crandall | first2=R. | last3=Ernvall | first3=R. | last4=Metsänkylä | first4=T. | title=Irregular Primes and Cyclotomic Invariants to Four Million | year=1993 | journal=Mathematics of Computation | volume=61 | issue=203 | pages=151–153 | url=http://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf | doi=10.2307/2152942| jstor=2152942 | bibcode=1993MaCom..61..151B | doi-access=free }} {{webarchive|url=https://archive.today/20210922005744/https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf|date=22 September 2021}}
{{Citation | last1=Clarke | first1=F. | last2=Jones | first2=C. | title=A Congruence for Factorials | year=2004 | journal=Bulletin of the London Mathematical Society | volume=36 | pages=553–558 | url=http://blms.oxfordjournals.org/content/36/4/553.full.pdf | doi=10.1112/S0024609304003194 | issue=4| s2cid=120202453 }} {{webarchive|url=https://www.webcitation.org/5vRE6GbVK?url=http://blms.oxfordjournals.org/content/36/4/553.full.pdf|date=2 January 2011}}
{{Citation | last1=Johnson | first1=W. | title=Irregular Primes and Cyclotomic Invariants | year=1975 | journal=Mathematics of Computation | volume=29 | issue=129 | pages=113–120 | url=http://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf | doi=10.2307/2005468| jstor=2005468 | doi-access=free }} {{webarchive|url=https://archive.today/20211228083543/https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf|date=28 December 2021}}
{{Citation | last1=McIntosh | first1=R. J. | title=On the converse of Wolstenholme's Theorem | year=1995 | journal=Acta Arithmetica | volume=71 | issue=4 | pages=381–389 | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf| doi=10.4064/aa-71-4-381-389 | doi-access=free }}
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{{Citation | author1-link=Paulo Ribenboim | last1=Ribenboim | first1=P. | title=The Little Book of Bigger Primes | location=New York | publisher=Springer-Verlag New York, Inc. | year=2004 | isbn=978-0-387-20169-6 |chapter=Chapter 2. How to Recognize Whether a Natural Number is a Prime}}
{{Citation | last1=Selfridge | first1=J. L. | last2=Pollack | first2=B. W. | title=Fermat's last theorem is true for any exponent up to 25,000 | year=1964 | journal=Notices of the American Mathematical Society | volume=11 | pages=97}}
{{Citation | last1=Trevisan | first1=V. | last2=Weber | first2=K. E. | title=Testing the Converse of Wolstenholme's Theorem | year=2001 | journal=Matemática Contemporânea | volume=21 | issue=16 | pages=275–286 | doi=10.21711/231766362001/rmc2116 | hdl=10183/448 | url=http://www.lume.ufrgs.br/bitstream/handle/10183/448/000317407.pdf?sequence=1}} {{webarchive|url=https://web.archive.org/web/20111006064608/http://www.lume.ufrgs.br/bitstream/handle/10183/448/000317407.pdf?sequence=1|date=6 October 2011}}
{{Citation | last1=Zhao | first1=J. | title=Bernoulli numbers, Wolstenholme's theorem, and p⁵ variations of Lucas' theorem | year=2007 | journal=Journal of Number Theory | volume=123 | pages=18–26 | doi=10.1016/j.jnt.2006.05.005 | s2cid=937685 | url=http://home.eckerd.edu/~zhaoj/research/ZhaoJNTBern.pdf| doi-access=free }}{{webarchive|url=https://web.archive.org/web/20100630160329/http://home.eckerd.edu/~zhaoj/research/ZhaoJNTBern.pdf|date=30 June 2010}}
{{Citation | last1=Zhao | first1=J. | title=Wolstenholme Type Theorem for Multiple Harmonic Sums | year=2008 | journal=International Journal of Number Theory | volume=4 | issue=1 | pages=73–106 | url=http://home.eckerd.edu/~zhaoj/research/ZhaoIJNT.pdf | doi=10.1142/s1793042108001146}}

External links

Caldwell, Chris K. [http://primes.utm.edu/glossary/xpage/Wolstenholme.html Wolstenholme prime] from The Prime Glossary
McIntosh, R. J. [http://www.loria.fr/~zimmerma/records/Wieferich.status Wolstenholme Search Status as of March 2004] e-mail to Paul Zimmermann
Bruck, R. [https://web.archive.org/web/20130208001700/http://imperator.usc.edu/~bruck/research/stirling Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients]
Conrad, K. [http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/padicharmonicsum.pdf The p-adic Growth of Harmonic Sums] interesting observation involving the two Wolstenholme primes

Category:Classes of prime numbers

Category:Unsolved problems in number theory