## Contributions to some conjectures on a ternary exponential Diophantine equation

### Volume 186 / 2018

Acta Arithmetica 186 (2018), 1-36
MSC: Primary 11D61; Secondary 11J86, 11D41.
DOI: 10.4064/aa8656-2-2018
Published online: 12 October 2018

#### Abstract

We consider the exponential Diophantine equation $a^x+b^y=c^z$ for given pairwise coprime positive integers $a$, $b$ and $c$. The case where both $x$ and $y$ are even is thoroughly studied in order to solve the equation for some infinite classes of triples $(a,b,c)$ which have been unassailable via the previously existing methods. It may be recognized from one of our results that a “quarter” of a major unsolved problem of Jeśmanowicz concerning primitive Pythagorean triples is almost solved.