I wonder if you could get away with a proof that is almost identical to induction. If you can prove that for n = 1 is true, and then consider what happens between the differences - that is, when you have n^3 - 7n + 3 and (n+1)^3 - 7(n+1) + 3 - is a multiple of three, then n^3 - 7n + 3 must be a multiple of three because you are adding multiple of threes to your first term...it's not strictly induction if you don't put on the extra comments and assuming it's true for n = k, no?