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

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

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

Solution 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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