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The equation formula here has no solution with x square

Published online by Cambridge University Press:  01 November 1999

Y. BUGEAUD ,
M. MIGNOTTE ,
Y. ROY  and
T. N. SHOREY
Y. BUGEAUD
Affiliation:
Université Louis Pasteur, UFR de Mathématiques, 7 rue René Descartes, 67084 Strasbourg, France
M. MIGNOTTE
Affiliation:
Université Louis Pasteur, UFR de Mathématiques, 7 rue René Descartes, 67084 Strasbourg, France
Y. ROY
Affiliation:
Université Louis Pasteur, UFR de Mathématiques, 7 rue René Descartes, 67084 Strasbourg, France
T. N. SHOREY
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
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Abstract

We study the Diophantine equation

formula here

in integers x, y > 1, n > 2 and q [ges ] 2. Without loss of generality, we suppose that q is prime.

Saradha and Shorey [SS] considered equation (1·1) with x = z2 and proved that it has no solutions in the cases z > 31 or z ∈ {2, 3, 4, 8, 9, 16, 27}.Notice that the proof for z ∈ {2, 4, 8, 16} is elementary, whereas for z ∈ {3, 9, 27} the proof uses a result of Darmon–Merel [DM] which is a generalization of Wiles' result on Fermat's theorem. The purpose of the present paper is to treat completely the remaining cases and to prove the following result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 3 , November 1999 , pp. 353 - 372
Copyright
© The Cambridge Philosophical Society 1999

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