A NOTE ON THE DIOPHANTINE EQUATION

Let (a, b, c)be a primitive Pythagorean triple satisfying {a}^{2} + {b}^{2} = {c}^{2} . In 1956, Jeśmanowicz conjectured that for any given positive integer nthe only solution of \mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} in positive integers is x= y= z= 2. In this paper, for the primitive Pythagorean triple (a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)with k= {2}^{s} for some...