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Jeśmanowicz’ Conjecture with Congruence Relations. II

Published online by Cambridge University Press:  20 November 2018

Yasutsugu Fujita
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan e-mail:
Takafumi Miyazaki
Department of Mathematics, College of Science and Technology, Nihon University 1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan e-mail:
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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 ${{b}_{0}}\,\equiv \,\in \,\,\,\left( \bmod \,a \right)$ with $\text{ }\!\!\varepsilon\!\!\text{ }\,\in \,\left\{ \pm 1 \right\}$ for certain positive divisors ${{b}_{0}}$ of $b$, then the Diophantine equation ${{a}^{x}}\,+\,{{b}^{y}}\,=\,{{c}^{z}}$ has only the positive solution $\left( x,\,y,\,z \right)\,=\,\left( 2,\,2,\,2 \right)$.

Research Article
Copyright © Canadian Mathematical Society 2014


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