Abstract
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| = 1.
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