Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita ¹   ; Takafumi Miyazaki ²

¹ Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei Narashino, Chiba, Japan
² Department of Mathematics and Information Sciences Tokyo Metropolitan University 1-1 Minami-Ohsawa Hachioji, Tokyo, Japan

Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 211-222

Résumé

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 $c \equiv -1 \pmod{a}$.

Détail

DOI : 10.4064/cm128-2-6

Keywords: relatively prime positive integers prove equiv pmod equiv pmod non negative integer diophantine equation has only positive solution holds equiv pmod