Geodesic
A Note on the Diophantine Equation x 2 + y 6 = ze , e ≥ 4
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 435-440

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-114-6
Mots-clés : 11D, diophantine equation 
Zelator, Konstantine. A Note on the Diophantine Equation x 2 + y 6 = ze , e ≥ 4. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 435-440. doi: 10.4153/CMB-2011-114-6
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     year = {2012},
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     doi = {10.4153/CMB-2011-114-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-114-6/}
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[1] Instructional conference and workshop on diophantine equations. Lorentz Center, Leiden University, The Netherlands. May 7–11, 2007 and May 14–16, 2007. Google Scholar

[2] [2] Sierpinski, W., Elementary theory of numbers. North-Holland Mathematical Library, 31. North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw, 1988, pp. 73–74, 78–79. Google Scholar

[3] [3] Pocklington, H. C., Some diophantine impossibilities. Proc. Cambridge Philos. Soc. 17(1914), 108–121. Google Scholar

[4] [4] Dickson, L. E., History of the theory of numbers. III. Quadratic and higher forms. AMS Chelsea Publishing, Providence, RI, 1992, Google Scholar

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