Find all rational solutions to $x^3 - y^2 = 2$.
The only integers solutions are $(3,\pm5)$:
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Glorfindel
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Christmas Bunny
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See Mordell-Bachet, rational solutions of $y^2=x^3-2$. – Dietrich Burde Dec 04 '13 at 18:29
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The equation $y^2 = x^3+k$, where $k \in \mathbb{Z}$ is called Mordell-Bachet equation. An excellent resource for this is Keith Conrad's article, where he discusses integer solutions. Your problem is on page $6$, theorem $3.4$. – Dec 04 '13 at 18:36
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See J. W. S. Cassels, The rational solutions of the diophantine equation $y^2 = x^3 - D.$ Acta Arithmetica, volume 82 (1950) pages 243-273. – Dietrich Burde Dec 04 '13 at 18:37