Problem 3

Let

P(x)

be a non-constant polynomial with integer coefficients such that

P(0)\neq0

. Let

a_1,a_2,a_3,\ldots

be an infinite sequence of integers such that

P(i-j)

divides

a_i-a_j

for all distinct positive integers

i,j

. Prove that the sequence

a_1,a_2,a_3,\ldots

must be constant, that is,

a_n

equals a constant

c

for every positive integer

n

Step 1 of 5: Bound the gcd of P(k) and k

Previous step Next step

\gcd(P(k),k)=\gcd(a_0,k)\le|a_0|,\qquad a_0=P(0)

Detailed analysis

Let $a_0=P(0)\neq0$ be the constant term of $P$ . Since $P$ has integer coefficients, $P(k)-a_0$ is divisible by $k$ , so $\gcd(P(k),k)=\gcd(a_0,k)\le|a_0|$ for every positive integer $k$ .