On Jeśmanowicz's conjecture (Q2742084)

Let \((a,b,c)\) be a primitive Pythagorean triple with \(2\mid b\). \textit{L. Jeśmanowicz} [Wiadom. Mat. 1, No. 2, 196-202 (1956; Zbl 0074.27205)] conjectured that the equation \(a^x+b^y=c^z\) has only the solution \((x,y,z)=(2,2,2)\). This problem is not yet resolved. In his paper the author proves that if the divisors of \(b\) satisfy some congruence conditions, then Jeśmanowicz' conjecture is true. Reviewer's remark: The lemma in the paper is an unchecked result due to Z. Cao.