Generalizations of classical results on Jeśmanowiczʼ conjecture concerning Pythagorean triples

Abstract In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple ( a , b , c ) satisfying a 2 + b 2 = c 2 , that the equation a x + b y = c z has the unique solution ( x , y , z ) = ( 2 , 2 , 2 ) in positive integers x , y and z . This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of classical well-known results on the conjecture. As a corollary we can verify that the conjecture is true if a − b = ± 1 .