MathDB

For a positive integer

n

we denote by

s(n)

the sum of the digits of

n

. Let

P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

be a polynomial, where

n \geqslant 2

and

a_i

is a positive integer for all

0 \leqslant i \leqslant n-1

. Could it be the case that, for all positive integers

k

s(k)

and

s(P(k))

have the same parity?

Problem Statement