Jeśmanowicz' conjecture on exponential diophantine equations

Takafumi Miyazaki

doi:10.7169/facm/1323705814

December 2011 Jeśmanowicz' conjecture on exponential diophantine equations

Takafumi Miyazaki

Funct. Approx. Comment. Math. 45(2): 207-229 (December 2011). DOI: 10.7169/facm/1323705814

Abstract

Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with even $b$. In 1956, L. Jeśmanowicz conjectured that the equation $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$ in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution $(x,y,z)$ with even $x,z$ then $x/2$ and $z/2$ are odd.

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Takafumi Miyazaki "Jeśmanowicz' conjecture on exponential diophantine equations," Functiones et Approximatio Commentarii Mathematici 45(2), 207-229, (December 2011). https://doi.org/10.7169/facm/1323705814

Published: December 2011

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