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://www.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://www.numdam.org/item/SB_2002-2003__45__1_0/

Bibliographie
Cité par

[1] M. F. Atiyah & I. G. Macdonald - Introduction to Commutative Algebra, Addison-Wesley, 1969. | MR

[2] A. Baker - “Linear forms in the logarithms of algebraic numbers I”, Mathematika 13 (1966), p. 204-216. | MR

[3] -, “Linear forms in the logarithms of algebraic numbers II”, Mathematika 14 (1967), p. 102-107.

[4] -, “Linear forms in the logarithms of algebraic numbers III”, Mathematika 14 (1967), p. 220-224.

[5] -, “Linear forms in the logarithms of algebraic numbers IV”, Mathematika 15 (1968), p. 204-216. | MR

[6] -, “Bounds for solutions of hyperelliptic equations”, Math. Proc. Cambridge Philos. Soc. 65 (1969), p. 439-444. | MR | Zbl

[7] A. Baker & G. Wüstholz - “Logarithmic forms and group varieties”, J. reine angew. Math. 442 (1993), p. 19-62. | EuDML | MR | Zbl

[8] C. D. Bennett, J. Blass, A. M. W. Glass, D. B. Meronk & R. P. Steiner - “Linear forms in the logarithms of three positive rational numbers”, J. Théor. Nombres Bordeaux 9 (1997), p. 97-136. | EuDML | Numdam | MR | Zbl

[9] Y. F. Bilu - “Catalan without logarithmic forms”, J. Théor. Nombres Bordeaux, to appear. | Numdam | Zbl

[10] J. Blass, A. M. W. Glass & T. W. O'Neil - “Catalan's conjecture and linear forms in logarithms”, Ulam Quart., accepted, but never appeared in print.

[11] Y. Bugeaud & G. Hanrot - “Un nouveau critère pour l'équation de Catalan”, Mathematika 47 (2000), p. 63-73. | MR | Zbl

[12] J. W. S. Cassels - “On the equation $a^{x} - b^{y} = 1$ . II”, Math. Proc. Cambridge Philos. Soc. 56 (1960), p. 97-103. | MR | Zbl

[13] E. Catalan - “Note extraite d'une lettre adressée à l'éditeur”, J. reine angew. Math. 27 (1844), p. 192. | EuDML | Zbl

[14] A. O. Gelfond - Transcendental and Algebraic Numbers, Moscow, 1952, (Russian); English transl.: New York, Dover, 1960. | MR | Zbl

[15] S. Hyyrö - “Über das Catalan'sche Problem”, Ann. Univ. Turku. Ser. A I 79 (1964), p. 3-10. | MR | Zbl

[16] K. Inkeri - “On Catalan's problem”, Acta Arith. 9 (1964), p. 285-290. | EuDML | MR | Zbl

[17] -, “On Catalan's conjecture”, J. Number Theory 34 (1990), p. 142-152. | Zbl

[18] C. Ko - “On the diophantine equation $x^{2} = y^{n} + 1$ , $x y \neq 0$ ”, Sci. Sinica 14 (1965), p. 457-460. | Zbl

[19] M. Langevin - “Quelques applications de nouveaux résultats de Van der Poorten”, in Sém. Delange-Pisot-Poitou (1975/1976), vol. 2, Paris, 1977. | EuDML | Numdam | Zbl

[20] M. Laurent, M. Mignotte & Y. Nesterenko - “Formes linéaires en deux logarithmes et déterminants d'interpolation”, J. Number Theory 55 (1995), p. 285-321. | MR | Zbl

[21] V. A. Lebesgue - “Sur l’impossibilité en nombres entiers de l’équation $x^{m} = y^{2} + 1$ ”, Nouv. Ann. Math. 9 (1850), p. 178-181. | EuDML

[22] E. Matveev - “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers I”, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998), p. 81-136, (Russian); English transl.: Izv. Math., 62 (1998), p. 723-772. | MR | Zbl

[23] -, “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II”, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), p. 125-180, (Russian); English transl.: Izv. Math., 64 (2000), p. 125-180. | MR | Zbl

[24] T. Metsänkylä - “Catalan's equation with a quadratic exponent”, C. R. Math. Rep. Acad. Sci. Canada 23 (2001), p. 28-32. | Zbl

[25] M. Mignotte - “Un critère élémentaire pour l'équation de Catalan”, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), p. 199-200. | MR | Zbl

[26] -, “Catalan's equation just before 2000”, in Number theory (Turku, 1999), de Gruyter, Berlin, 2001, p. 247-254. | MR | Zbl

[27] M. Mignotte & Y. Roy - “Catalan’s equation has no new solutions with either exponent less than $10651$ ”, Experimental Math. 4 (1995), p. 259-268. | EuDML | MR | Zbl

[28] -, “Minorations pour l'équation de Catalan”, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), p. 377-380. | MR | Zbl

[29] P. Mihăilescu - “A class number free criterion for Catalan's conjecture”, J. Number Theory 99 (2003), p. 225-231. | Zbl

[30] -, “Primary cyclotomic units and a proof of Catalan's conjecture”, J. reine angew. Math., to appear. | MR | Zbl

[31] -, “On the class group of cyclotomic extensions in the presence of a solution to Catalan's equation”, a manuscript. | Zbl

[32] T. W. O'Neil - “Improved upper bounds on the exponents in Catalan's equation”, a manuscript, 1995.

[33] J.-C. Puchta - “On a criterion for Catalan's conjecture”, Ramanujan J. 5 (2001), p. 405-407. | MR | Zbl

[34] P. Ribenboim - Catalan's Conjecture, Academic Press, Boston, 1994. | MR | Zbl

[35] W. Schwarz - “A note on Catalan's equation”, Acta Arith. 72 (1995), p. 277-279. | EuDML | MR | Zbl

[36] F. Thaine - “On the ideal class groups of real abelian number fields”, Ann. of Math. 128 (1988), p. 1-18. | MR | Zbl

[37] R. Tijdeman - “On the equation of Catalan”, Acta Arith. 29 (1976), p. 197-209. | EuDML | MR | Zbl

[38] M. Waldschmidt - “Minorations de combinaisons linéaires de logarithmes de nombres algébriques”, Canad. J. Math. 45 (1993), p. 176-224. | MR | Zbl

[39] L. Washington - Introduction to cyclotomic fields, 2nd 'ed., Graduate Texts in Math., vol. 83, Springer, New York, 1997. | MR | Zbl

[40] G. Wüstholz ('ed.) - A Panorama of Number Theory or The View from Baker's Garden, Cambridge University Press, 2002. | MR