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Un nouveau critère pour l'équation de Catalan

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

Yann Bugeaud  and
Guillaume Hanrot
Yann Bugeaud
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, F-67084 Strasbourg, France. E-mail: bugeaud@math.u-strasbg.fr
Guillaume Hanrot
Affiliation:
Projet POLKA, INRIA Lorraine, 615, rue du Jardin Botanique, B.P. 101, F-54602 Villers-lès-Nancy Cedex, France. E-mail: guillaume.hanrot@loria.fr
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Abstract

A new criterion on Catalan's equation is proved by elementary means

This shows, without appealing either to the theory of linear forms in logarithms, or to any computation, that (C) has no solution (x, y, p, q) with min {p, q}≤41, except (3,2, 2, 3).

Résumé

On démontre de manière élémentaire un nouveau critère pour l'équation de Catalan

II permet de prouver, sans faire appel à la théorie des formes linéaires de logarithmes ni au moindre calcul, qu'outre (3, 2, 2, 3), toute éventuelle solution (x, y, p, q) de (C) vérifie min {p, q}≥43.

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 63 - 73
Copyright
Copyright © University College London 2000

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References

1.André, Y.. G-functions and Geometry. Vieweg (Braunschweig 1989).CrossRefGoogle Scholar
2.Baker, A.. Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc., 65 (1969), 439–444.CrossRefGoogle Scholar
3.Bilu, Y.. Solving superelliptic Diophantine equations by the method of Gelfond-Baker, preprint 94–09, Mathématiques Stochastiques, Univ. Bordeaux 2 (1994).Google Scholar
4.Bilu, Y. et Hanrot, G.. Solving superelliptic Diophantine equations by Baker's method. Compositio Math., 112 (1998), 273–312.CrossRefGoogle Scholar
5.Bugeaud, Y.. Bounds for the solutions of superelliptic equations. Compositio Math., 107 (1997), 187–219.CrossRefGoogle Scholar
6.Cassels, J. W. S.. On the equation ax – by = 1, 11, Proc. Cambridge Philos. Society, 56 (1960), 97–103.CrossRefGoogle Scholar
7.Ernvall, R. et Metsänkylä, T.. On the p-divisibility of Fermat quotients. Math. Comp., 66 (1997), 1353–1365.CrossRefGoogle Scholar
8.Fung, G., Granville, A. et Williams, H. C.. Computation of the First Factor of the Class Number of Cyclotomic Fields. J. Number Theory, 42 (1992), 297–312.CrossRefGoogle Scholar
9.Hyyrö, S.. Über das Catalansche Problem. Ann. Univ. Turku Ser. AI 79 (1964), 3–10.Google Scholar
10.Inkeri, K.. On Catalan's problem. Acta Arith., 9 (1964), 285–290.CrossRefGoogle Scholar
11.Inkeri, K.. On Catalan's conjecture. J. Number Th., 34 (1990), 142–152.CrossRefGoogle Scholar
12.Chao, Ko. On the diophantine equation x2 = yn + 1, xy≠0, Sci. Sinica, 14 (1965), 457–460.Google Scholar
13.Langevin, M.. Quelques applications de nouveaux résultats de van der Poorten, Sém. Delange-Pisot-Poitou (1977/1978), Paris, Exp. 4, 7 pages.Google Scholar
14.Laurent, M., Mignotte, M. et Nesterenko, Y.. Formes lineaires en deux logarithmes et déterminants d'interpolation. J. Number Th., 55 (1995), 285–321.CrossRefGoogle Scholar
15.Lebesgue, V. A.. Sur l'impossibilite en nombres entiers de l'equation xm = y2 + 1, Nouv. Ann. Math., 9 (1850), 178–181.Google Scholar
16.Lehmer, D. H.. Prime factors of cyclotomic class numbers. Math. Comp., 31 (1977), 599–607.CrossRefGoogle Scholar
17.Lepistö, T.. On the growth of the first factor of the class number of the prime cyclotomic field. Ann. Acad. Sci. Fenn. Ser. Al Math., 577 (1974), 21 pages.Google Scholar
18.Le, Maohua. A note on the integer solutions of hyperelliptic equations. Colloq. Math., 68 (1995), 171–177.CrossRefGoogle Scholar
19.Ljunggren, W.. Noen Setninger om ubestemte likninger av formen (xn - 1)/(x - 1) = yq, Norsk. Mat. Tidsskr., 25 (1943), 17–20.Google Scholar
20.Mignotte, M.. Sur l'équation de Catalan, C. R. Acad. Sci. Paris, 314, (1992), 165–168.Google Scholar
21.Mignotte, M.. Un critère élémentaire pour l'equation de Catalan. C. R. Math. Rep. Acad. Sci. Canada, 15 (1993), 199–200.Google Scholar
22.Mignotte, M.. Sur l'equation de Catalan (II). Theoretical Computer Science, 123 (1994), 145–149.CrossRefGoogle Scholar
23.Mignotte, M.. A criterion on Catalan's equation. J. Numb. Th., 52 (1995), 280–283.CrossRefGoogle Scholar
24.Mignotte, M. et Roy, Y.. Catalan's equation has no new solution with either exponent less than 10651. Experimental Mathematics, 4 (1995), 259–268.CrossRefGoogle Scholar
25.Mignotte, M. et Roy, Y.. Minorations pour l'equation de Catalan. C. R. Acad. Sci. Paris, 324 (1997), 377–380.CrossRefGoogle Scholar
26.Mignotte, M. et Roy, Y.. Lower Bounds for Catalan's Equation. The Ramanujan J., 1 (1997), 351–356.CrossRefGoogle Scholar
27.Nagell, T.. Des équations indéterminees x2 + x + 1 = yn et x2 + x + 1 = 3yn. Nordsk. Mat. Forenings Skr. (1), 2 (1920), 14 pages.Google Scholar
28.Nagell, T.. Note sur l'equation indéterminee (xn - 1)/(x - 1) = yq. Norsk. Mat. Tidsskir., 2 (1920), 75–78.Google Scholar
29.Ribenboim, P.. Catalan's Conjecture. Academic Press (Boston 1994).Google Scholar
30.Runge, C.. Über ganzzahlige Losungen von Gleichungen zwischen zwei Veränderlichen. J. reine angew. Math., 100 (1887), 425–435.CrossRefGoogle Scholar
31.Schwarz, W.. A note on Catalan's equation. Acta Arith., 72 (1995), 277–279.CrossRefGoogle Scholar
32.Siegel, C. L.. Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys.-math. Kl., 1 (1929), 70 pages.Google Scholar
33.Tijdeman, R.. On the equation of Catalan. Acta Arith., 29 (1976), 197–209.CrossRefGoogle Scholar
34.Walsh, P. G.. A quantitative version of Runge's theorem on diophantine equations. Acta Arith., 62 (1992), 157–172.CrossRefGoogle Scholar
35.Washington, L. C.. Introduction to cyclotomic fields. Springer-Verlag (New York 1982).CrossRefGoogle Scholar