On Je smanowicz' Conjecture Concerning Pythagorean Triples

Let (a;b;c) be a primitive Pythagorean triple. Jesmanowicz conjectured in 1956 that for any positive integer n, the Diophantine equation (an) x + (bn) y = (cn) z has only the positive integer solution (x;y;z) = (2;2;2). Let p 3 (mod 4) be a prime and s be some positive integer. In the paper, we show that the conjecture is true when (a;b;c) = (4p 2s 1;4p s ;4p 2s + 1) and certain divisibility conditions are...