Generalizations of classical results on Jeśmanowiczʼ conjecture concerning Pythagorean triples

In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple (a,b,c) satisfying a2+b2=c2, that the equation ax+by=cz 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.