Jeśmanowicz' conjecture on exponential Diophantine equations (Q651798)

Jeśmanowicz' conjecture is the following statement: If \(a\), \(b\), \(c\) are coprime positive integers such that \(a^2+b^2=c^2\) with even \(b\), then the exponential equation \(a^x+b^y=c^z\) has the only solution \((x,y,z)=(2,2,2)\) in positive integers. This paper contains various new results on this conjecture. In particular, the author proves that if the previous equation has a solution \((x,y,z)\) with \(x\) and \(z\) even then \(x/2\) and \(z/2\) are odd.