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?
Hint:
Show that for $x$ large enough: $$(x+2)^3<x^3+8x^2-6x+8<(x+3)^3$$ Then you are left with very few cases to check.