Generalizations of classical results on Jeśmanowicz’ conjecture concerning Pythagorean triples

In 1956, Leon Jeśmanowicz conjectured that, for any primitive Pythagorean triple (a,b,c) with a2+b2 = c2, the equation ax+by = cz in positive integers has only the solution (x,y,z) = (2,2,2). There are some classical and celebrated results on this conjecture. In this paper, we broadly generalize many of them. As a corollary, we can verify that the conjecture is true when |b−a| =...