Let ( a , b , c ) be a primitive Pythagorean triple such that a 2 + b 2 = c 2 with 2| b . In 1956, L. Jesmanowicz conjectured that, for any positive integer n , the equation ( a n ) x + ( b n ) y = ( c n ) z has only the positive solution ( x , y , z ) = (2, 2, 2). In 1959, Lu Wenduan claimed that if n = 1 and ( a , b , c ) = (4 k 2 − 1, 4 k , 4 k 2 + 1), then the conjecture is true. Denote by P ( n ) the product of the prime factors of n . In this paper, we prove that the conjecture is true for n > 1, ( a , b , c ) = (4 k 2 − 1, 4 k , 4 k 2 + 1), under some conditions.
Deng et al. (Thu,) studied this question.