Editing Wilson's theorem (section)

==History==
The theorem was first stated by [[Ibn al-Haytham]] {{circa|1000 AD}}.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> [[Edward Waring]] announced the theorem in 1770 without proving it, crediting his student [[John Wilson (mathematician and judge)|John Wilson]] for the discovery.<ref>Edward Waring, ''Meditationes Algebraicae'' (Cambridge, England:  1770), page 218 (in Latin).  In the third (1782) edition of Waring's ''Meditationes Algebraicae'', Wilson's theorem appears as problem 5 on [https://books.google.com/books?id=1MNbAAAAQAAJ&pg=PA380 page 380].  On that page, Waring states:  "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger."  (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)</ref> [[Joseph Louis Lagrange|Lagrange]] gave the first proof in 1771.<ref>Joseph Louis Lagrange, [https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA125 "Demonstration d'un théorème nouveau concernant les nombres premiers"] (Proof of a new theorem concerning prime numbers), ''Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres'' (Berlin), vol. 2, pages 125–137 (1771).</ref>  There is evidence that [[Gottfried Leibniz|Leibniz]] was also aware of the result a century earlier, but never published it.<ref>[[Giovanni Vacca (mathematician)|Giovanni Vacca]] (1899) "Sui manoscritti inediti di Leibniz" (On unpublished manuscripts of Leibniz),
''Bollettino di bibliografia e storia delle scienze matematiche'' ... (Bulletin of the bibliography and history of mathematics), vol. 2, pages 113–116; see [https://books.google.com/books?id=vqwSAQAAMAAJ&pg=PA114 page 114] (in Italian).  Vacca quotes from Leibniz's mathematical manuscripts kept at the Royal Public Library in Hanover (Germany), vol. 3 B, bundle 11, page 10:  <blockquote>''Original'' :  Inoltre egli intravide anche il teorema di Wilson, come risulta dall'enunciato seguente:<br/>"Productus continuorum usque ad numerum qui antepraecedit datum divisus per datum relinquit 1 (vel complementum ad unum?) si datus sit primitivus.  Si datus sit derivativus relinquet numerum qui cum dato habeat communem mensuram unitate majorem."<br/>Egli non giunse pero a dimostrarlo.</blockquote><blockquote>''Translation'' :  In addition, he [Leibniz] also glimpsed Wilson's theorem, as shown in the following statement:<br/>"The product of all integers preceding the given integer, when divided by the given integer, leaves 1 (or the complement of 1?) if the given integer be prime.  If the given integer be composite, it leaves a number which has a common factor with the given integer [which is] greater than one."<br/>However, he didn't succeed in proving it.</blockquote> See also:  Giuseppe Peano, ed., ''Formulaire de mathématiques'', vol. 2, no. 3, [https://archive.org/details/formulairedemat02peangoog/page/n231 page 85] (1897).</ref>