Jeśmanowicz' conjecture with congruence relations

Volume 128 / 2012

Yasutsugu Fujita, Takafumi Miyazaki Colloquium Mathematicum 128 (2012), 211-222 MSC: Primary 11D61; Secondary 11D09. DOI: 10.4064/cm128-2-6

Abstract

Let $a,b$ and $c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}$. We prove that if $b \equiv 0 \pmod{2^{r}}$ and $b \equiv \pm 2^{r} \pmod{a}$ for some non-negative integer $r$, then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the positive solution $(x,y,z)=(2,2,2)$. We also show that the same holds if $c \equiv -1 \pmod{a}$.

Authors

  • Yasutsugu FujitaDepartment of Mathematics
    College of Industrial Technology
    Nihon University
    2-11-1 Shin-ei
    Narashino, Chiba, Japan
    e-mail
  • Takafumi MiyazakiDepartment of Mathematics and Information Sciences
    Tokyo Metropolitan University
    1-1 Minami-Ohsawa
    Hachioji, Tokyo, Japan
    e-mail

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