Ogled vira Wilsonovo praštevilo

{{Infopolje številsko zaporedje
| named_after           = [[John Wilson (matematik)|John Wilson]]
| publication_year      = 1938<ref>{{Cite journal | doi = 10.2307/1968791 | last = Lehmer | first = Emma | authorlink = Emma Lehmer | title = On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson | journal = [[Annals of Mathematics]] | volume = 39 | issue = 2 | pages = 350–360 | date = April 1938 | url = http://gradelle.educanet2.ch/christian.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf | accessdate = 8 March 2011| jstor = 1968791 }}</ref>
| author                = [[Emma Lehmer]]
| terms_number          = 3
| first_terms           = [[5 (število)|5]], [[13 (število)|13]], [[563 (število)|563]]
| largest_known_term    = [[563 (število)|563]]
| OEIS                  = A007540
| OEIS_name             = Wilsonova praštevila: praštevila p, da velja (p-1)! == -1 (mod p^2)
}}
'''Wilsonovo praštevilo''', poimenovano po angleškem matematiku [[John Wilson (matematik)|Johnu Wilsonu]], je vsako [[praštevilo]] ''p'', da ''p''<sup>2</sup> [[Delitelj|deli]] (''p''&nbsp;&#x2212;&nbsp;1)!&nbsp;+&nbsp;1, kjer označuje »!« [[Fakulteta (funkcija)|faktorialno funkcijo]]. Ime in zaporedje izhajata iz [[Wilsonov izrek|Wilsonovega izreka]], ki pravi, da vsako praštevilo ''p'' deli (''p''&nbsp;&#x2212;&nbsp;1)!&nbsp;+&nbsp;1.

Edina znana Wilsonova praštevila so [[5 (število)|5]], [[13 (število)|13]] in [[500 (število)|563]] {{OEIS|id=A007540}}. Znano je tudi, da če obstajajo še kakšna druga Wilsonova praštevila, morajo biti večja od 2{{E|13}}.<ref name="Search2">[[arxiv:1209.3436|A Search for Wilson primes]] Retrieved on November 2, 2012.</ref> [[Domneva|Domneva se]], da obstaja neskončno mnogo Wilsonovih praštevil, število takšnih praštevil na intervalu [''x'',&nbsp;''y''], pa je približno log(log(''y'')/log(''x'')).<ref>[http://primes.utm.edu/glossary/page.php?sort=WilsonPrime The Prime Glossary: Wilson prime]</ref>

Izvedlo se je več računalniških iskanj Wilsonovih praštevil.<ref>{{Navedi splet|last=McIntosh|first=R.|authorlink=Richard McIntosh|title=WILSON STATUS (Feb. 1999)|website=E-Mail to [[Paul Zimmermann (mathematician)|Paul Zimmermann]]|date=9 March 2004|url=http://www.loria.fr/~zimmerma/records/Wieferich.status|accessdate=6 June 2011}}</ref><ref>''A search for Wieferich and Wilson primes'', p 443</ref><ref>{{Navedi knjigo|last=Ribenboim|first=P.|authorlink=Paulo Ribenboim|last2=Keller|first2=W.|title=Die Welt der Primzahlen: Geheimnisse und Rekorde|publisher=Springer|year=2006|location=Berlin Heidelberg New York|page=241|language=de|url=https://books.google.com/?id=-nEM9ZVr4CsC&pg=PA248&dq=die+welt+der+primzahlen+rodenkirch#v=onepage&q&f=false|isbn=978-3-540-34283-0}}</ref>
[[Ibercivis|Ibercivisovo]] [[porazdeljeno izračunavanje]] vključuje iskanje Wilsonovih praštevil.<ref>{{Navedi splet |url=http://www.ibercivis.net/index.php?module=public&section=channels&action=view&id_channel=2&id_subchannel=138 |title=Ibercivis site |accessdate=2020-09-02 |archive-date=2012-06-20 |archive-url=https://web.archive.org/web/20120620210249/http://www.ibercivis.net/index.php?module=public&section=channels&action=view&id_channel=2&id_subchannel=138 |url-status=dead }}</ref> Drugo iskanje upravlja forum [[Veliko spletno iskanje Mersennovih praštevil]].<ref>[http://www.mersenneforum.org/showthread.php?t=16028 Distributed search for Wilson primes] {{Webarchive|url=https://web.archive.org/web/20200318161652/http://www.mersenneforum.org/showthread.php?t=16028 |date=2020-03-18 }} (at mersenneforum.org)</ref>

== Posplošitve ==

=== Wilsonova praštevila reda {{Mvar|n}} ===
Wilsonov izrek se lahko v splošnem izrazu <math>(n-1)!(p-n)!\equiv(-1)^n\ \bmod p</math> za vsako celo število <math>n \ge 1</math> in praštevilo <math>p \ge n</math>. Posplošena Wilsonova praštevila reda {{Mvar|n}} so praštevila {{Mvar|p}}, da <math>p^2</math> deli <math>(n-1)!(p-n)! - (-1)^n</math>.

Domnevano je, da obstaja za vsako naravno število {{Mvar|n}} neskončno mnogo Wilsonovih praštevil reda {{Mvar|n}}.
{| class="wikitable"
!<math>n</math>
!praštevilo <math>p</math>, da <math>p^2</math> deli <math>(n-1)!(p-n)! - (-1)^n</math> (preverjeno do 1000000)
!Zaporedje
[[OEIS]]
|-
|1
|5, 13, 563, ...
|{{OEIS link|A007540}}
|-
|2
|2, 3, 11, 107, 4931, ...
|{{OEIS link|A079853}}
|-
|3
|7, ...
|
|-
|4
|10429, ...
|
|-
|5
|5, 7, 47, ...
|
|-
|6
|11, ...
|
|-
|7
|17, ...
|
|-
|8
|...
|
|-
|9
|541, ...
|
|-
|10
|11, 1109, ...
|
|-
|11
|17, 2713, ...
|
|-
|12
|...
|
|-
|13
|13, ...
|
|-
|14
|...
|
|-
|15
|349, 41341, ...
|
|-
|16
|31, ...
|
|-
|17
|61, 251, 479, ...
|{{OEIS link|A152413}}
|-
|18
|13151527, ...
|
|-
|19
|71, 621629, ...
|
|-
|20
|59, 499, 43223, 214009, ...
|
|-
|21
|217369, ...
|
|-
|22
|...
|
|-
|23
|...
|
|-
|24
|47, 3163, ...
|
|-
|25
|...
|
|-
|26
|97579, ...
|
|-
|27
|53, ...
|
|-
|28
|347, 739399, ...
|
|-
|29
|...
|
|-
|30
|137, 1109, 5179, ...
|
|}
Najmanj posplošena Wilsonova praštevila reda ''n'' so:

: 5, 2, 7, 10429, 5, 11, 17, ... (Naslednji člen > 1.4×10<sup>7</sup>) {{OEIS|id=A128666}}

== Glej tudi ==

* [[Wieferichovo praštevilo]]
* [[Wall-Sun-Sunovo praštevilo]]
* [[Wolstenholmovo praštevilo]]
* [[PrimeGrid]]
* [[tabela kongruenc]]

== Sklici ==

{{sklici|3}}

== Viri ==

{{refbegin|2}}
* {{Navedi časopis|last=Beeger|first=N. G. W. H.|year=1913–1914|title=Quelques remarques sur les congruences ''r''<sup>''p''−1</sup> ≡&nbsp;1 (mod&nbsp;''p''<sup>2</sup>) et (''p''&nbsp;−&nbsp;1!)&nbsp;≡&nbsp;−1 (mod p<sup>2</sup>)|journal=[[Messenger of Mathematics|The Messenger of Mathematics]]|volume=43|pages=72–84}}
* {{Navedi časopis|last=Goldberg|first=Karl|year=1953|title=A table of Wilson quotients and the third Wilson prime|journal=[[J. London Math. Soc.]]|volume=28|issue=2|pages=252–256|doi=10.1112/jlms/s1-28.2.252}}
* {{Navedi knjigo|title=The new book of prime number records|url=https://archive.org/details/newbookprimenumb00ribe_678|first=Paulo|last=Ribenboim|authorlink=Paulo Ribenboim|publisher=[[Springer-Verlag]]|year=1996|isbn=978-0-387-94457-9|pages=[https://archive.org/details/newbookprimenumb00ribe_678/page/n365 346]}}
* {{Navedi časopis|last=Crandall|first=Richard E.|last2=Dilcher|first2=Karl|last3=Pomerance|first3=Carl|year=1997|title=A search for Wieferich and Wilson primes|journal=Math. Comput.|volume=66|issue=217|pages=433–449|doi=10.1090/S0025-5718-97-00791-6|doi-access=free}}
* {{Navedi knjigo|title=Prime Numbers: A Computational Perspective|url=https://archive.org/details/primenumberscomp0000cran|first=Richard E.|last=Crandall|first2=Carl|last2=Pomerance|publisher=Springer-Verlag|year=2001|page=[https://archive.org/details/primenumberscomp0000cran/page/29 29]|isbn=978-0-387-94777-8}}
* {{Navedi časopis|last=Pearson|first=Erna H.|year=1963|title=On the Congruences (''p''&nbsp;−&nbsp;1)! ≡&nbsp;−1 and 2<sup>''p''−1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>)|url=http://www.ams.org/journals/mcom/1963-17-082/S0025-5718-1963-0159780-0/S0025-5718-1963-0159780-0.pdf|journal=Math. Comput.|volume=17|pages=194–195}}
{{refend}}

== Zunanje povezave ==

* [http://primes.utm.edu/glossary/page.php?sort=WilsonPrime The Prime Glossary: Wilson prime]
* {{MathWorld|urlname=WilsonPrime|title=Wilson prime}}
* [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wilson primes]

[[Kategorija:Nerešeni matematični problemi]]
[[Kategorija:Razredi praštevil]]