For a polynomial P with coefficients in Z, it's trivial that P(n) is integer if n is. The converse is not true. There are lots of polynomials that always give integer answers to integer questions, but whose coefficients are not integers. For example, n(n+1)/2 = (1/2)n + (1/2)n^2 always takes integer values for integer n, even though the coefficients aren't integers. Is there a way to quickly eyeball a polynomial in general to see if it is Z -> Z? If the coefficients are rational, one can find K = the LCM of the denominators, multiply through by K, and test it for all the integers from 0 to K-1 to see if the result is always divisible by K. But I am hoping there is a simpler way. If any of the coefficients are irrational, my intuition is that the polynomial is never Z->Z, but I haven't been able to think of an easy proof.