MathDB

Integer polynomial commutes with sum of digits

Source: 2016 IMO Shortlist N1

July 19, 2017

number theoryIMO Shortlistpolynomialfunctional equationsum of digits

Problem Statement

For any positive integer

k

, denote the sum of digits of

k

in its decimal representation by

S(k)

. Find all polynomials

P(x)

with integer coefficients such that for any positive integer

n \geq 2016

, the integer

P(n)

is positive and

S(P(n)) = P(S(n)).

Proposed by Warut Suksompong, Thailand