Catalan's conjecture

Catalan's conjecture
[L'hypothèse de Catalan]

Bilu, Yuri F.

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 909, pp. 1-26.

Résumé
Abstract

Le sujet de cet exposé est le travail récent de Mihăilescu, qui a démontré que l’équation $x^{p} - y^{q} = 1$ n’a pas de solutions en entiers non-zero $x, y$ et premiers impairs $p, q$ . En combinaison avec les résultats de Lebesgue (1850) et Ko Chao (1865), ceci implique l’hypothèse célèbre de Catalan (1843) : l’équation $x^{u} - y^{v} = 1$ n’a pas de solutions en entiers $x, y > 0$ et $u, v > 1$ sauf $3^{2} - 2^{3} = 1$ . 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.

The subject of the talk is the recent work of Mihăilescu, who proved that the equation $x^{p} - y^{q} = 1$ has no solutions in non-zero integers $x, y$ and odd primes $p, q$ . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated conjecture of Catalan (1843): the only solution to $x^{u} - y^{v} = 1$ in integers $x, y > 0$ and $u, v > 1$ is $3^{2} - 2^{3} = 1$ . 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.

Zbl

Classification : 11D61, 11R18, 11J86, 11R27, 11R33, 11Y50
Keywords: unités cyclotomiques, paires de Wieferich
Mot clés : cyclotomic units, Wieferich's pairs

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     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/}
}

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Bilu, Yuri F. Catalan's conjecture, dans Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 909, pp. 1-26. http://archive.numdam.org/item/SB_2002-2003__45__1_0/

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