Editing Bézout's identity (section)

== History and attribution ==

The French mathematician [[Étienne Bézout]] (1730–1783) proved this identity for polynomials.<ref>{{cite book |author=Bézout, É.|author-link=Étienne Bézout|url=https://archive.org/details/bub_gb_RDEVAAAAQAAJ |title=Théorie générale des équations algébriques |place=Paris, France |publisher=Ph.-D. Pierres |year=1779}}</ref> The statement for integers can be found already in the work of an earlier French mathematician, [[Claude Gaspard Bachet de Méziriac]] (1581–1638).<ref>
{{cite book | last=Tignol | first=Jean-Pierre | author-link=Jean-Pierre Tignol | title=Galois' Theory of Algebraic Equations | publisher=World Scientific| location=Singapore | year=2001 | isbn=981-02-4541-6}}</ref><ref>{{cite book|author=Claude Gaspard Bachet (sieur de Méziriac)|title=Problèmes plaisants & délectables qui se font par les nombres|edition=2nd|location=Lyons, France|publisher=Pierre Rigaud & Associates|year=1624|pages= 18–33|url=http://www.bsb-muenchen-digital.de/~web/web1008/bsb10081407/images/index.html?digID=bsb10081407&pimage=38&v=100&nav=0&l=de}}  On these pages, Bachet proves (without equations) "Proposition XVIII.  Deux nombres premiers entre eux estant donnez, treuver le moindre multiple de chascun d’iceux, surpassant de l'unité un multiple de l'autre."  (Given two numbers [which are] relatively prime, find the lowest multiple of each of them [such that] one multiple exceeds the other by unity (1).)  This problem (namely, {{math|1=''ax'' − ''by'' = 1}}) is a special case of Bézout's equation and was used by Bachet to solve the problems appearing on pages 199 ff.</ref><ref>See also: {{cite journal|date=February 2009|author=Maarten Bullynck |title=Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany |doi=10.1016/j.hm.2008.08.009|journal=Historia Mathematica|volume=36|issue=1|pages=48–72| url=http://hal.inria.fr/docs/00/66/32/92/PDF/Gauss_Modular_Oct2008.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://hal.inria.fr/docs/00/66/32/92/PDF/Gauss_Modular_Oct2008.pdf |archive-date=2022-10-09 |url-status=live| doi-access=free}}</ref>
[[Andrew Granville]] traced the association of Bézout's name with the identity to [[Nicolas Bourbaki|Bourbaki]], arguing that it is a misattribution since the identity is implicit in [[Euclid's Elements|Euclid's ''Elements'']].<ref>{{cite arXiv |last=Granville |first=Andrew |date=2024 |title=It is not "Bézout's identity" |arxiv=2406.15642 |class=math.HO}}</ref>