Jeśmanowicz' conjecture and related equations
Volume 184 / 2018
Acta Arithmetica 184 (2018), 37-49
MSC: Primary 11D61; Secondary 11D41.
DOI: 10.4064/aa170508-17-9
Published online: 12 April 2018
Abstract
Jeśmanowicz conjectured that the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,y, z)=(2,2,2)$, where $m$ and $n$ are positive integers with $m \gt n$, $\gcd(m, n)=1$ and $m\not\equiv n\pmod{2}$. In this paper, we first improve the result of Miyazaki and Terai (2015). Let $a, b$ and $c$ be coprime integers with $a+b=c$. We also obtain some results on the exponential equation $(an)^x+(bn)^y=(cn)^z$.
