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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


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." Funct. Approx. Comment. Math. 45 (2) 207 - 229, December 2011.


Published: December 2011
First available in Project Euclid: 12 December 2011

zbMATH: 1266.11064
MathSciNet: MR2895155
Digital Object Identifier: 10.7169/facm/1323705814

Primary: 11D61
Secondary: 11D41 , 11J86

Keywords: exponential diophantine equations , generalized Fermat equations , lower bounds for linear forms in logarithms of algebraic numbers , Pythagorean triples

Rights: Copyright © 2011 Adam Mickiewicz University

Vol.45 • No. 2 • December 2011
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