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

Open Access article
 Printed: Jun 2002
  • O. Kihel
  • C. Levesque
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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 $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles' deep machinery.
Keywords: Diophantine equations Diophantine equations
MSC Classifications: 11D41 show english descriptions Higher degree equations; Fermat's equation 11D41 - Higher degree equations; Fermat's equation

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