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