Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with even $b$. In 1956, L. Jeśmanowicz conjectured that the equation $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$ in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution $(x,y,z)$ with even $x,z$ then $x/2$ and $z/2$ are odd.
"Jeśmanowicz' conjecture on exponential diophantine equations." Funct. Approx. Comment. Math. 45 (2) 207 - 229, December 2011. https://doi.org/10.7169/facm/1323705814