A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only solution \((x,y,z)=(2,2,2)\). this connection we call \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. 1999 M.-H. Le gave necessary conditions existence exceptional solutions which were refined recently by H. Yang R.-Q. Fu. paper give unified simple proof theorem Le-Yang-Fu. Next in case \(v=2,\ u\) is an odd prime. As application show truth conjecture all prime values \(u < 100\).