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

  Published:2011-06-08
 Printed: Jun 2012
  • Konstantine Zelator,
    Department of Mathematics and Computer Science, Rhode Island College, Providence, RI 02908, USA
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Abstract

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.
Keywords: diophantine equation diophantine equation
MSC Classifications: 11D show english descriptions unknown classification 11D 11D - unknown classification 11D
 

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