The subject of the talk is the recent work of Mihăilescu, who proved that the equation has no solutions in non-zero integers and odd primes . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated conjecture of Catalan (1843): the only solution to in integers and is . Before the work of Mihăilescu the most definitive result on Catalan's problem was due to Tijdeman (1976), who proved that the solutions of Catalan's equation are bounded by an absolute effective constant.
Le sujet de cet exposé est le travail récent de Mihăilescu, qui a démontré que l’équation n’a pas de solutions en entiers non-zero et premiers impairs . En combinaison avec les résultats de Lebesgue (1850) et Ko Chao (1865), ceci implique l’hypothèse célèbre de Catalan (1843) : l’équation n’a pas de solutions en entiers et sauf . Avant ce travail de Mihăilescu, le résultat le plus définitif sur le problème de Catalan était celui de Tijdeman (1976), qui a démontré que les solutions de l'équation de Catalan sont bornées par une constante absolue effective.
Keywords: unités cyclotomiques, paires de Wieferich
Mot clés : cyclotomic units, Wieferich's pairs
@incollection{SB_2002-2003__45__1_0, author = {Bilu, Yuri F.}, title = {Catalan's conjecture}, booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923}, series = {Ast\'erisque}, note = {talk:909}, pages = {1--26}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {294}, year = {2004}, zbl = {1094.11014}, language = {en}, url = {http://archive.numdam.org/item/SB_2002-2003__45__1_0/} }
TY - CHAP AU - Bilu, Yuri F. TI - Catalan's conjecture BT - Séminaire Bourbaki : volume 2002/2003, exposés 909-923 AU - Collectif T3 - Astérisque N1 - talk:909 PY - 2004 SP - 1 EP - 26 IS - 294 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://archive.numdam.org/item/SB_2002-2003__45__1_0/ LA - en ID - SB_2002-2003__45__1_0 ER -
%0 Book Section %A Bilu, Yuri F. %T Catalan's conjecture %B Séminaire Bourbaki : volume 2002/2003, exposés 909-923 %A Collectif %S Astérisque %Z talk:909 %D 2004 %P 1-26 %N 294 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://archive.numdam.org/item/SB_2002-2003__45__1_0/ %G en %F SB_2002-2003__45__1_0
Bilu, Yuri F. Catalan's conjecture, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 909, pp. 1-26. http://archive.numdam.org/item/SB_2002-2003__45__1_0/
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