2002 IMO Shortlist Problems/N1
What is the smallest positive integer such that there exist integers with
Observe that . On the other hand, each cube is congruent to 0, 1, or -1 modulo 9. So a sum of at most three cubes modulo 9 must among none of which are congruent to 4. Therefore .
To show that 4 is the minimum value of , note that