Prove an equation has no integer solutions...

Question

I know that ${x^3} - 8{y^3} = 12$ has no integer solutions but how can I prove it? If I had to sit down with someone and convince them (at least, fairly) rigorously that it has no integer solutions. How would I do it? Any help is appreciated.

Answer 1

Suppose $ (x,y) $ is a solution. Then $ 2 | 8y^3 + 12 = x^3 $ implying $ 2 | x $. Writing $ x = 2k $ gives $ 8(k^3 - y^3) = 12 $, a contradiction as $ 8 $ doesn't divide $ 12 $.