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Find all integer solutions to $x^2+4=y^3$.

Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?

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marked as duplicate by Zander, O.L., Julian Kuelshammer, Lord_Farin, azimut May 26 '13 at 12:25

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You forgot the tag elliptic-curves. Regards. –  awllower May 5 '13 at 1:29
Yes. Good point. :) –  user70520 May 5 '13 at 1:33

1 Answer 1

These are examples of Mordell's equation. The only solutions are $(\pm 2,2)$ and $(\pm 11,5)$ to equation $x^2+4 = y^3$. The same problem is discussed in theorem $3.3$ on page $6$ here. The article, by Keith Conrad, is a wonderful article on solutions to certain Mordell's equation and is worth reading.

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Indeed a good note: his method for dealing with such curves is quite consistent, and tends more to algebraic methods than elliptic curves. And I think it is relatively difficult to adapt the elliptic viewpoint here. –  awllower May 5 '13 at 1:49
@awllower Yes. Only a basic calculus background is needed to understand, all the equations solved in the article. –  user17762 May 5 '13 at 1:51
A calculus background?!? There's no calculus used at that link. Rather, one should know basic number theory (e.g., when $2 \bmod p$ is a square and how factoring and norms are related in ${\mathbf Z}[i]$). –  KCd May 5 '13 at 4:40
@KCD Maybe that means calculus to user 17762? Like calculus and number-theort are both too elementary to be distinguished? Just saying, anyway.:D –  awllower May 5 '13 at 9:12

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