Let n be an integer. Can both n + 3 and n2 + 3 be perfect cubes?
If n + 3 and n2 + 3 are both perfect cubes then their product must also be a perfect cube.
So consider (n + 3)(n2 + 3) = n3 + 3n2 + 3n + 9 = (n + 1)3 + 8.
This can be a perfect cube only if it is 8 more than another perfect cube, namely (n + 1)3.
The only pairs of perfect cubes that differ by 8 are (−8, 0) and (0, 8).
So we must have
For neither of these solutions is n2 + 3 a perfect cube.
Therefore, if n is an integer, n + 3 and n2 + 3 cannot both be perfect cubes.
For what integers n is 18(n2 + 3) a perfect cube?
Source: Inspired by Mathematical Miniatures, by Svetoslav Savchev and Titu Andreescu. See Coffee Break 1.