integer solutions of the equation $a^3-5b^2$=2?

Question

Has the equation

$a^3-5b^2$=2

any integer solutions ?

With brute force, I checked that a must be greater than 10^9.

Answer 1

This equation has no integral solutions; in fact, it has no rational ones. In order to confirm this, start by substituting $a=\frac{1}{5}x, b=\frac{1}{25}y$, which yields $$x^3 - 250 = y^2$$

It's clear that rational solutions of the original equation correspond to rational solutions of the new one and vice versa. Equation of this kind defines so-called elliptic curve and there are known methods for calculating the integer/rational points on such curves.

As an example, PARI/GP shows the following (the vector $[0,0,0,0,-250]$ describes the curve):

? e=ellinit([0,0,0,0,-250]);
? ellgenerators(e)
[]
? elltors(e)
[1, [], []]

The first part of the output (= empty vector) shows that there are no rational points on the curve of infinite order. The second part shows that the torsion subgroup of the curve is trivial too, so there are no non-trivial rational points of finite order either. Put together, there are no rational points on this curve, so there can't be any integer solutions to the original equation.