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This question already has an answer here:

Find all rational solutions to $x^3 - y^2 = 2$.

The only integers solutions are $(3,\pm5)$:

http://mathforum.org/library/drmath/view/51569.html

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marked as duplicate by Dietrich Burde, user17762, Daniel Fischer, egreg, user61527 Dec 4 '13 at 18:59

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  • $\begingroup$ See Mordell-Bachet, rational solutions of $y^2=x^3-2$. $\endgroup$ – Dietrich Burde Dec 4 '13 at 18:29
  • $\begingroup$ 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 $3.4$. $\endgroup$ – user17762 Dec 4 '13 at 18:36
  • $\begingroup$ 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. $\endgroup$ – Dietrich Burde Dec 4 '13 at 18:37