Solutions to Mordell's equation $x^2 = y^3 + 8$

Question

It seems to be known that all the integral solutions to this equation are (0, -2) ($\pm$ 3, 1), ($\pm$ 4, 2), ($\pm$ 312, 46). Yet I can't seem to find any proofs that these are all the solutions.

Using the (semi)-unique factorization in $Z[\sqrt{8}]$, we get $x + \sqrt{8} = \alpha \omega (a + b\sqrt{8})^3$, where $a, b$ are relatively prime integers, $a$ is odd, $\omega \in \{1, 3 + \sqrt{8}, 3 - \sqrt{8} \}$ and $\alpha \in \{ \sqrt{8}, 4 + \sqrt{8}, 1\}$. I was able to take care of the case $\omega = 1$ easily, but have had problems with the other two cases; the algebra gets tricky and messy.

I've also tried playing with the factorization $y^3 + 8 = (y+2)(y^2 - 2y + 4)$ and have concluded that the only primes that divide both factors on the right hand side are 2 and 3, but that's it.

Any help would be appreciated.

Answer 1

The purpose of this answer is to make it possible to view every future question about Mordell equation $y^2=x^3+n$ a duplicate. See J. Gebel, A. Petho and H. G. Zimmer, ‘On Mordell’s equations’, Compos. Math. 110 (1998) no. 3, 335–367 where integral solutions of all such equations are found for $n<10^4$.

Here is a quote from this article by Bennett:

There are alternative approaches for finding the integral points on a given model of an elliptic curve. The most commonly used currently proceeds via appeal to lower bounds for linear forms in elliptic logarithms, the idea for which dates back to work of Lang [27] and Zagier [38] (though the bounds required to make such arguments explicit are found in work of David and of Hirata-Kohno; see, for example, [6]). Using these bounds, Gebel et al. [16], Smart [33] and Stroeker and Tzanakis [34] obtained, independently, a ‘practical’ method to find integral points on elliptic curves. Applying this method, in 1998, Gebel et al. [17] solved equation (1.1) for all integers $|k| < 10^4$ and partially extended the computation to $|k| < 10^5$ .