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A NOTE ON THE DIOPHANTINE EQUATION $\mathop{(na)}\nolimits ^{x} + \mathop{(nb)}\nolimits ^{y} = \mathop{(nc)}\nolimits ^{z} $

Published online by Cambridge University Press:  07 August 2013

Department of Applied Mathematics, Hainan University, Haikon, Hainan 570228, PR China email
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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 $n$ the 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 positive integer $s\geq 0$, we prove the conjecture when $n\gt 1$ and certain divisibility conditions are satisfied.

MSC classification

Research Article
Copyright ©2013 Australian Mathematical Publishing Association Inc. 


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