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A Note on the Diophantine Equation x2 + y6 = ze, e ≥ 4
Published online by Cambridge University Press: 20 November 2018
Abstract
We consider the diophantine equation ${{x}^{2}}\,+\,{{y}^{6}}\,=\,{{z}^{e}},\,e\,\le \,4$. 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.
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- Copyright © Canadian Mathematical Society 2012