On the diophantine equation $x^n+y^n=2^{\alpha}pz^2$

C. R. Math. Rep. Acad. Sci. Canada Vol. 28, (1), 2006 pp. 6–11
(Received: 2006-03-01 )

Michael A. Bennett, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: bennett@math.ubc.ca

Jamie Mulholland, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2; email: jmulholl@math.ubc.ca

Abstract/Résumé:

We show, if \(p\) is prime, that the equation \(x^n+y^n=2pz^2\) has no solutions in coprime integers \(x\), \(y\) and \(z\) with \(|xy|>1\) and prime \(n>p^{27p^2}\), and, if \(p\ne7\), the equation \(x^n+y^n=pz^2\) has no solutions in coprime integers \(x\), \(y\) and \(z\) with \(|xy|>1\), \(z\) even and prime \(n>p^{3p^2}\).

Nous montrons que, si \(p\) est premier, l’équation \(x^n+y^n=2pz^2\) n’a pas de solution parmi les nombres entiers copremiers \(x\), \(y\), \(z\), avec \(|xy| > 1\) et \(n>p^{27p^2}\) premier. Nous montrons aussi que, si \(p\ne7\), l’équation \(x^n+y^n=pz^2\) n’a pas de solution parmi les nombres entiers copremiers \(x\), \(y\), \(z\), avec \(|xy| >1\), \(z\) pair, et \(n>p^{3p^2}\) premier.

Keywords:

AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

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