A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$

http://dx.doi.org/10.4153/CMB-2011-114-6
Canad. Math. Bull. 55(2012), 435-440

  Published:2011-06-08
 Printed: Jun 2012

We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$. We show that, when $e$ is a multiple of $4$ or $6$, this equation has no solutions in positive integers with $x$ and $y$ relatively prime. As a corollary, we show that there exists no primitive Pythagorean triangle one of whose leglengths is a perfect cube, while the hypotenuse length is an integer square.