Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita¹ ; Takafumi Miyazaki²

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 $b\equiv 0(\mathrm{mod}{2}^r)$ and $b\equiv \pm 2^r(\mathrm{mod}a)$ 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 $c\equiv -1(\mathrm{mod}a)$.

Détail

DOI : 10.4064/cm128-2-6

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