Well, an unknown polynomial of degree 3n has 3n+1 unknown coefficients. At first blush, therefore, it seems like any positive n can be made to work.

/*******************************/

For instance, let n = 1. Then P(0) = 2, P(1) = 1, P(2) = 0.

P(x) = (x-2)(ax(x-1)-1) for any a

Then P(4) = 2*(12a-1) = 730so a = 30.5

P(x) = (x-2)(30.5x(x-1)-1) satisfies the problem requirements

/*********************************/

But I expect that any higher n can be made to work also.

blackjack

flooble's webmaster puzzle