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Jeśmanowicz’ Conjecture with Congruence Relations. II
Published online by Cambridge University Press: 20 November 2018
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
${{b}_{0}}\,\equiv \,\in \,\,\,\left( \bmod \,a \right)$ 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
$\left( x,\,y,\,z \right)\,=\,\left( 2,\,2,\,2 \right)$.
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- Research Article
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- Copyright © Canadian Mathematical Society 2014
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