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| {{Infobox integer sequence
| | They call me Ned and therefore i think this might quite good when you say this kind of. To keep bees is mysterious cure he loves most. Since she was 18 she's been working as being a production and planning officer but soon her husband and her will start their own company. My family lives in Nebraska. Go to her website to learn more: http://raleighx6.blog.com/2014/06/16/la-fibra-de-vidrio-un-componente-util-con-formidable-capacidad-para-el-reciclaje/<br><br>Feel free to visit my web site [http://raleighx6.blog.com/2014/06/16/la-fibra-de-vidrio-un-componente-util-con-formidable-capacidad-para-el-reciclaje/ resina epoxi] |
| | named_after = [[John Wilson (Mathematician)|John Wilson]]
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| | publication_year = 1938<ref>{{Cite journal | doi = 10.2307/1968791 | last = Lehmer | first = E. | 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}}</ref>
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| | author = [[Emma Lehmer]]
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| | terms_number = 3
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| | first_terms = [[5 (number)|5]], [[13 (number)|13]], [[563 (number)|563]]
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| | largest_known_term = [[563 (number)|563]]
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| | OEIS = A007540
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| }}
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| A '''Wilson prime''', named after [[English people|English]] mathematician [[John Wilson (mathematician)|John Wilson]], is a [[prime number]] ''p'' such that ''p''<sup>2</sup> divides (''p'' − 1)! + 1, where "!" denotes the [[factorial function]]; compare this with [[Wilson's theorem]], which states that every prime ''p'' divides (''p'' − 1)! + 1.
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| The only known Wilson primes are [[5 (number)|5]], [[13 (number)|13]], and 563 {{OEIS|id=A007540}}; if any others exist, they must be greater than 2{{e|13}}.<ref name="Search">[http://arxiv.org/pdf/1209.3436v2.pdf A Search for Wilson primes] Retrieved on November 2, 2012.</ref> It has been [[conjecture]]d that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [''x'', ''y''] is about log(log(''y'')/log(''x'')).<ref>[http://primes.utm.edu/glossary/page.php?sort=WilsonPrime The Prime Glossary: Wilson prime]</ref>
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| Several computer searches have been done in the hope of finding new Wilson primes.<ref>{{Cite web | last = McIntosh | first = R. | authorlink = Richard McIntosh | title = WILSON STATUS (Feb. 1999) | work = E-Mail to 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>{{Cite book | last = Ribenboim | first = P. | authorlink = Paulo Ribenboim | coauthors = Keller, W. | title = Die Welt der Primzahlen: Geheimnisse und Rekorde | publisher = Springer | date = 2006 | location = Berlin Heidelberg New York | page = 241 | language = German | url = http://books.google.de/books?id=-nEM9ZVr4CsC&pg=PA248&dq=die+welt+der+primzahlen+rodenkirch&hl=de&ei=LbLsTfG8G8XdsgamnLXnCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q&f=false | isbn = 3-540-34283-4}}</ref>
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| The [[Ibercivis]] [[distributed computing]] project includes a search for Wilson primes.<ref>[http://www.ibercivis.net/index.php?module=public§ion=channels&action=view&id_channel=2&id_subchannel=138 Ibercivis site]</ref> Another search is coordinated at the mersenneforum.<ref>[http://www.mersenneforum.org/showthread.php?t=16028 Distributed search for Wilson primes] (at mersenneforum.org)</ref>
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| ==Generalizations==
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| ===Near-Wilson primes===
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| A prime p satisfying the congruence (p − 1)! ≡ − 1 + ''Bp'' (mod ''p''<sup>2</sup>) with small |''B''| can be called a '''near-Wilson prime'''. Near-Wilson primes with ''B'' = 0 represent Wilson primes. The following table lists all such primes with |''B''| ≤ 100 from 10<sup>6</sup> up to 4{{e|11}}:<ref name="Search"/>
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| {| class="wikitable collapsible collapsed" style="width:30%;border:0px;text-align:right;"
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| |-
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| ! p !! B
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| |-
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| | 1282279 || +20
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| |-
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| | 1306817 || −30
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| |-
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| | 1308491 || −55
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| |-
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| | 1433813 || −32
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| |-
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| | 1638347 || −45
| |
| |-
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| | 1640147 || −88
| |
| |-
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| | 1647931 || +14
| |
| |-
| |
| | 1666403 || +99
| |
| |-
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| | 1750901 || +34
| |
| |-
| |
| | 1851953 || −50
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| |-
| |
| | 2031053 || −18
| |
| |-
| |
| | 2278343 || +21
| |
| |-
| |
| | 2313083 || +15
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| |-
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| | 2695933 || −73
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| |-
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| | 3640753 || +69
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| |-
| |
| | 3677071 || −32
| |
| |-
| |
| | 3764437 || −99
| |
| |-
| |
| | 3958621 || +75
| |
| |-
| |
| | 5062469 || +39
| |
| |-
| |
| | 5063803 || +40
| |
| |-
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| | 6331519 || +91
| |
| |-
| |
| | 6706067 || +45
| |
| |-
| |
| | 7392257 || +40
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| |-
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| | 8315831 || +3
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| |-
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| | 8871167 || −85
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| |-
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| | 9278443 || −75
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| |-
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| | 9615329 || +27
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| |-
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| | 9756727 || +23
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| |-
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| | 10746881 || −7
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| |-
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| | 11465149 || −62
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| |-
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| | 11512541 || −26
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| |-
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| | 11892977 || −7
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| |-
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| | 12632117 || −27
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| |-
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| | 12893203 || −53
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| |-
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| | 14296621 || +2
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| |-
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| | 16711069 || +95
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| |-
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| | 16738091 || +58
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| |-
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| | 17879887 || +63
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| |-
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| | 19344553 || −93
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| |-
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| | 19365641 || +75
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| |-
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| | 20951477 || +25
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| |-
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| | 20972977 || +58
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| |-
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| | 21561013 || −90
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| |-
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| | 23818681 || +23
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| |-
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| | 27783521 || −51
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| |-
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| | 27812887 || +21
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| |-
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| | 29085907 || +9
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| |-
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| | 29327513 || +13
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| |-
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| | 30959321 || +24
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| |-
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| | 33187157 || +60
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| |-
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| | 33968041 || +12
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| |-
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| | 39198017 || −7
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| |-
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| | 45920923 || −63
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| |-
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| | 51802061 || +4
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| |-
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| | 53188379 || −54
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| |-
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| | 56151923 || −1
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| |-
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| | 57526411 || −66
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| |-
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| | 64197799 || +13
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| |-
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| | 72818227 || −27
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| |-
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| | 87467099 || −2
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| |-
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| | 91926437 || −32
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| |-
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| | 92191909 || +94
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| |-
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| | 93445061 || −30
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| |-
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| | 93559087 || −3
| |
| |-
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| | 94510219 || −69
| |
| |-
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| | 101710369 || −70
| |
| |-
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| | 111310567 || +22
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| |-
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| | 117385529 || −43
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| |-
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| | 176779259 || +56
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| |-
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| | 212911781 || −92
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| |-
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| | 216331463 || −36
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| |-
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| | 253512533 || +25
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| |-
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| | 282361201 || +24
| |
| |-
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| | 327357841 || −62
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| |-
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| | 411237857 || −84
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| |-
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| | 479163953 || −50
| |
| |-
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| | 757362197 || −28
| |
| |-
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| | 824846833 || +60
| |
| |-
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| | 866006431 || −81
| |
| |-
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| | 1227886151 || −51
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| |-
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| | 1527857939 || −19
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| |-
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| | 1636804231 || +64
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| |-
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| | 1686290297 || +18
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| |-
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| | 1767839071 || +8
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| |-
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| | 1913042311 || −65
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| |-
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| | 1987272877 || +5
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| |-
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| | 2100839597 || −34
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| |-
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| | 2312420701 || −78
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| |-
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| | 2476913683 || +94
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| |-
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| | 3542985241 || −74
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| |-
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| | 4036677373 || −5
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| |-
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| | 4271431471 || +83
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| |-
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| | 4296847931 || +41
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| |-
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| | 5087988391 || +51
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| |-
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| | 5127702389 || +50
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| |-
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| | 7973760941 || +76
| |
| |-
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| | 9965682053 || −18
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| |-
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| | 10242692519 || −97
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| |-
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| | 11355061259 || −45
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| |-
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| | 11774118061 || −1
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| |-
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| | 12896325149 || +86
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| |-
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| | 13286279999 || +52
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| |-
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| | 20042556601 || +27
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| |-
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| | 21950810731 || +93
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| |-
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| | 23607097193 || +97
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| |-
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| | 24664241321 || +46
| |
| |-
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| | 28737804211 || −58
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| |-
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| | 35525054743 || +26
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| |-
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| | 41659815553 || +55
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| |-
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| | 42647052491 || +10
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| |-
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| | 44034466379 || +39
| |
| |-
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| | 60373446719 || −48
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| |-
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| | 64643245189 || −21
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| |-
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| | 66966581777 || +91
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| |-
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| | 67133912011 || +9
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| |-
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| | 80248324571 || +46
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| |-
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| | 80908082573 || −20
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| |-
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| | 100660783343 || +87
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| |-
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| | 112825721339 || +70
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| |-
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| | 231939720421 || +41
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| |-
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| | 258818504023 || +4
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| |-
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| | 260584487287 || −52
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| |-
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| | 265784418461 || −78
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| |-
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| | 298114694431 || +82
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| |-
| |
| |}
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| ===Wilson numbers===
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| A ''Wilson number'' is an integer ''m'' such that ''W''(''m'') ≡ 0 (mod ''m''), where ''W''(''m'') denotes the [[Wilson quotient]] (ie. <math>\tfrac{(m-1)!+1}{m}</math>) {{OEIS|id=A157250}}. If ''m'' is prime, then ''m'' is a Wilson prime. There are 12 Wilson numbers up to 5{{e|8}}.<ref>{{cite journal | doi=10.1090/S0025-5718-98-00951-X | author=Takashi Agoh | coauthors=Karl Dilcher, Ladislav Skula | title=Wilson quotients for composite moduli | journal=Math. Comput. | volume=67 | issue=222 | pages=843–861 | year=1998 | url=http://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00951-X/S0025-5718-98-00951-X.pdf}}</ref>
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| == See also ==
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| * [[Wieferich prime]]
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| * [[Wall–Sun–Sun prime]]
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| * [[Wolstenholme prime]]
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| *[[PrimeGrid]]
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| * [[Table of congruences]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite journal | author=N. G. W. H. Beeger | title=Quelques remarques sur les congruences ''r''<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>) et (''p'' − 1!) ≡ −1 (mod p<sup>2</sup>) | journal=[[Messenger of Mathematics|The Messenger of Mathematics]] | volume=43 | pages=72–84 | year=1913–1914}}
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| * {{cite journal | author=Karl Goldberg | title=A table of Wilson quotients and the third Wilson prime | journal=[[J. London Math. Soc.]]| volume=28 | issue=2 | pages=252–256 | year=1953 | doi=10.1112/jlms/s1-28.2.252 }}
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| * {{cite book | title=The new book of prime number records | author=Paulo Ribenboim | authorlink=Paulo Ribenboim | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94457-5 | pages=346 }}
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| * {{cite journal | author=Richard E. Crandall | coauthors=Karl Dilcher, Carl Pomerance | title=A search for Wieferich and Wilson primes | journal=Math. Comput. | volume=66 | issue=217 | pages=433–449 | year=1997 | doi=10.1090/S0025-5718-97-00791-6 }}
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| * {{cite book | title=Prime Numbers: A Computational Perspective | author=Richard E. Crandall | coauthors=Carl Pomerance | publisher=Springer-Verlag | year=2001 | page=29 | isbn=0-387-94777-9 }}
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| * {{cite journal | author=Erna H. Pearson | title=On the Congruences (''p'' − 1)! ≡ −1 and 2<sup>''p''−1</sup> ≡ 1 (mod ''p''<sup>2</sup>) | journal=Math. Comput. | volume=17 | pages=194–195 | year=1963 | url=http://www.ams.org/journals/mcom/1963-17-082/S0025-5718-1963-0159780-0/S0025-5718-1963-0159780-0.pdf}}
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| == External links ==
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| * [http://primes.utm.edu/glossary/page.php?sort=WilsonPrime The Prime Glossary: Wilson prime]
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| * {{MathWorld|urlname=WilsonPrime|title=Wilson prime}}
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| * [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wilson primes]
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| * [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.6544&rep=rep1&type=pdf Wilson Quotients for composite moduli]
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| * [http://gradelle.educanet2.ch/christian.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson]
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| {{Prime number classes|state=collapsed}}
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| [[Category:Classes of prime numbers]]
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| [[Category:Factorial and binomial topics]]
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They call me Ned and therefore i think this might quite good when you say this kind of. To keep bees is mysterious cure he loves most. Since she was 18 she's been working as being a production and planning officer but soon her husband and her will start their own company. My family lives in Nebraska. Go to her website to learn more: http://raleighx6.blog.com/2014/06/16/la-fibra-de-vidrio-un-componente-util-con-formidable-capacidad-para-el-reciclaje/
Feel free to visit my web site resina epoxi