A remark on Jeśmanowicz' conjecture
Let \(a,b,c\) be a primitive Pythagorean triple such that \(a=r^2-s^2\), \(b=2rs\), \(c=r^2+s^2\), where \(r,s\) are positive integers satisfying \(r>s\), \(\gcd(r,s)=1\) and \(2\mid rs\). \textit{L. Jeśmanowicz} [Wiad. Mat. 1, 196--202 (1956; Zbl 0074.27205)] conjectured that the equation \((*)\) \(a^x+b^y=c^z\), \(x,y,z\in\mathbb N\), has only the solution \((x,y,z)=(2,2,2)\). This conjecture has been proved to be true in many special cases. But, in general, the problem is not solved as yet. Using Baker's method, the reviewer [Proc. Japan Acad., Ser. A 72, No. 5, 97--98 (1996; Zbl 0876.11013)] proved that if \(2\parallel r\), \(s\equiv 3\pmod 4\) and \(r\geq 81s\), then \((*)\) has only the solution \((x,y,z)=(2,2,2)\). In this paper, using some elementary methods, the author shows that the condition \(r\geq 81s\) can be eliminated from the above-mentioned result for \(s=3,7,11\) and 15.
scientific article; zbMATH DE number 952356