number theory - Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

The only solution I have found is $x=0$.

I have tried proving it by congruences and have had no success. I don't know how to prove it. I have even tried algebraic manipulation and have got

$$x(x^2 + 8x-6) = (y-2)(y^2 + 2y+4)$$

Could you please give me some ideas on how to proceed?

Answer

Hint:

Show that for $x$ large enough:
$$(x+2)^3Then you are left with very few cases to check.