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# Jeśmanowicz' Conjecture with Congruence Relations. II

Published:2014-04-28
Printed: Sep 2014
• Yasutsugu Fujita,
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
• Takafumi Miyazaki,
Department of Mathematics, College of Science and Technology, Nihon University 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
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## Abstract

Let $a,b$ and $c$ be primitive Pythagorean numbers such that $a^{2}+b^{2}=c^{2}$ with $b$ even. In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$ with $\epsilon \in \{\pm1\}$ for certain positive divisors $b_0$ of $b$, then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the positive solution $(x,y,z)=(2,2,2)$.
 Keywords: exponential Diophantine equations, Pythagorean triples, Pell equations
 MSC Classifications: 11D61 - Exponential equations 11D09 - Quadratic and bilinear equations

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