How to solve an equation $ x^3-y^2=20 $

Question

X and y in this equation are both integer. One can see that pair x=6 and y=14, as well as x=6 and y=-14 are both solutions of this equation. It's easy to prove also that if x>6 entails $$ x=[y^{2/3}]+1$$ where as a [m] I gonna denominate an integer part of number m - max integer number which doesnt exceed m. For example, [8]=8, [1.6]=1, [-1.6]=-2. But I wonder if this goes long way and dont have any good idea about this equation. If your solution of this equation contains some interesting theory, links are appretiated.

Answer 1

The equation $$y^2 = x^3-20$$ is an example of a Mordell equation (cf. https://en.wikipedia.org/wiki/Mordell_curve). You can find solutions for many $k$ in the range $-1000 \le k \le 1000$ at http://tnt.math.se.tmu.ac.jp/simath/MORDELL/.

Answer 2

$y^2 = x^3-20\tag{1}$
Using online Magma calculator as follows.

$E$:=EllipticCurve($[0, 0, 0, 0, -20]$);
IntegralPoints($E$);

It says that all integral points are $ [ (6 : -14 : 1) ] $.
Hence all integral points are $(x,y)=(6,\pm14).$