MathDB

Call a polynomial

f(x)

excellent if its coefficients are all in [0, 1) and

f(x)

is an integer for all integers

x

. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most

n

, in terms of

n

. c) Find the minimum

n\ge3

for which there exists an excellent polynomial of the form

\frac{1}{n!}x^n+g(x)

, where

g(x)

is a polynomial of degree at most

n-3

C4