Jeśmanowicz' conjecture with congruence relations
Volume 128 / 2012
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 $c \equiv -1 \pmod{a}$.
