# Mathematical ReportsComptes rendus mathématiques

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

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

PDF(click to download): On the diophantine equation $x^n+y^n=2^{alpha}pz^2$