Prove the following statement:

There is an infinite number of integer pairs (n,n+1) such that each of the integers cannot be represented by a sum of 3 integer cubes.

This is fairly easy to solve. Cubes mod 9 are congruent to 0, 1, or 8. There is no way to choose with repeats three of those congruences to add up to a number congruent to 4 or 5 mod 9. So all integer pairs (9x+4,9x+5) require at least four cubes.

It turns out we had this before with

3 cubes? - not always.