Problem 2

For a polynomial

P

and a positive integer

n

, define

P_n

as the number of positive integer pairs

(a,b)

such that

a<b\le n

and

|P(a)|-|P(b)|

is divisible by

n

. Determine all polynomials

P

with integer coefficients such that

P_n\le2021

for all positive integers

n

Step 4 of 6: The linear coefficient is ±1

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P(x)=cx+d,\quad |c|=1

Detailed analysis

Write $P(x)=cx+d$ with $c\ne0$ . If $|c|\ge2$ , then taking $n=|c|\ge2$ , $P(1)=c+d\equiv d\pmod c$ and $P(2)=2c+d\equiv d\pmod c$ are congruent modulo $n=|c|$ , contradicting the lemma applied with $y=1,z=2$ . Hence $|c|=1$ .