Problem #1 - Tackling the Diophantine

"Mathematics is the queen of the sciences and number theory is the queen of mathematics." - G.H. Hardy

Even though I might not concentrate in Number Theory during my upcoming PhD years, but what can I say, she is still my 'first love'. So let's get started with a Diophantine Equation !

I found this problem in my latest homework about Ring Theory especially interesting :

Find the integer solutions (x,y) of

\[x^3-2 = y^2\]

There is an advanced solution revolving around the ring $ \mathbb{Z}[\sqrt{-2}] $, which I just found recently (you don't wanna hear where I solved it), but the challenge is can you give a solution using Elementary Number Theory ?

For your information, there are exactly 2 pairs of integer solutions for this equation.

You're up !