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Jeśmanowicz' Conjecture with Congruence Relations. II

  • 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 exponential Diophantine equations, Pythagorean triples, Pell equations
MSC Classifications: 11D61, 11D09 show english descriptions Exponential equations
Quadratic and bilinear equations
11D61 - Exponential equations
11D09 - Quadratic and bilinear equations
 

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