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On a Few Diophantine Equations Related to Fermat’s Last Theorem

Published online by Cambridge University Press:  20 November 2018

O. Kihel  and
C. Levesque
O. Kihel
Affiliation:
Dept. of Math. and Comp. Sc., University of Lethbridge, Lethbridge, Alberta, T1K 3M4
C. Levesque
Affiliation:
Dép. Mathématiques et CICMA, Université Laval, Québec, G1K 7P4
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Abstract

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We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations ${{X}^{4}}\,-\,4{{Y}^{4}}\,=\,{{Z}^{p}},\,{{X}^{4}}\,+\,4{{Y}^{p}}\,=\,{{Z}^{2}}$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles’ deep machinery.

Keywords

11D41Diophantine equations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 2 , 01 June 2002 , pp. 247 - 256
Copyright
Copyright © Canadian Mathematical Society 2002

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