Does there exist any positive integer base N ≥ 8
such that 2007 is a perfect cube?
(In reply to another solution
The only thing to make sure of is if the inequality 7 + 2*N^3 < (N+1)^3 holds for N.
A litttle algebra gives N^3 - 3N^2 - 3N + 6 < 0. This inequality is true for N in the range 1.16 to 3.36 and if N is less than -1.53.
The inequality is never true for any N>=8, and is not usable in a proof.