# Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

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

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

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