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SUR L’ÉQUATION DIOPHANTIENNE (xn − 1)/(x − 1) = yq, III

Published online by Cambridge University Press:  13 February 2002

YANN BUGEAUD ,
GUILLAUME HANROT  and
MAURICE MIGNOTTE
YANN BUGEAUD
Affiliation:
Université Louis Pasteur, U.F.R. de Mathématiques 7, rue René Descartes 67084, Strasbourg, Francebugeaud@math.u-strasbg.fr
GUILLAUME HANROT
Affiliation:
Projet POLKA INRIA Lorraine & LORIA, 615, rue du Jardin Botanique, 54602 Villers-lès-Nancy Cedex, Franceguillaume.hanrot@loria.fr
MAURICE MIGNOTTE
Affiliation:
Université Louis Pasteur, U.F.R. de Mathématiques 7, rue René Descartes 67084, Strasbourg, Francemignotte@math.u-strasbg.fr
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Abstract

In this paper we study the diophantine equation of the title, which was first introduced by Nagell and Ljunggren during the first half of the twentieth century. We describe a method which allows us, on the one hand when n is fixed, to obtain an upper bound for q, and on the other hand when n and q are fixed, to obtain upper bounds for x and y which are far sharper than those derived from the theory of linear forms in logarithms. We also show how these bounds can be used even when they seem too large for a straightforward enumeration of the remaining possible values of x. By combining all these techniques, we are able to solve the equation in many cases, including the case when n has a prime divisor less than 13, or the case when n has a prime divisor which is less than or equal to 23 and distinct from q.

2000 Mathematical Subject Classification: primary 11D41; secondary 11J86, 11Y50.

Keywords

diophantine equationssuperelliptic equations
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 84 , Issue 1 , March 2002 , pp. 59 - 78
Copyright
2002 London Mathematical Society

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