MathDB

Eventually Periodic Residues of Polynomial Indicate Rational Coefficients

Source: 2023 China Team Selection Test Round 2 Day 3 Problem 20

April 3, 2023

polynomialnumber theory

Problem Statement

Let

a,b,d

be integers such that

\left|a\right| \geqslant 2

d \geqslant 0

and

b \geqslant \left( \left|a\right| + 1\right)^{d + 1}

. For a real coefficient polynomial

f

of degree

d

and integer

n

, let

r_n

denote the residue of

\left[ f(n) \cdot a^n \right]

mod

b

. If

\left \{ r_n \right \}

is eventually periodic, prove that all the coefficients of

f

are rational.