On Jeśmanowicz' conjecture concerning primitive Pythagorean triples. II
The Jeśmanowicz conjecture states that for given \(m,n\in\mathbb{Z}\) with \(m>n>0,\gcd m,n=1\) and \(m\not\equiv n\pmod{2}\) the equation \[ (m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z \] has the only solution \((x,y,z)=(2,2,2)\). In this paper the authors prove that the Jeśmanowicz conjecture is true under the following additional assumption: \(n\equiv 2\), \(m>72n\), at least one of the following three conditions is fulfilled -- \(n/2\) is a power of an odd prime; -- \(n/2\) has no prime factors congruent to 1 modulo 8; -- \(n/2\) is a square. As a corollary the authors show that this result implies that the Jeśmanowicz conjecture is true for every \(n<100\) for which \(n/2\) is odd. Part I, cf. the second author [J. Number Theory 141, 316-323 (2014; Zbl 1309.11030)].
scientific article; zbMATH DE number 6699218