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 2 of 5: Choose k with no small common factor

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k\text{ prime},\ k>|a_0|\implies \gcd(P(k),k)=1

Detailed analysis

For $k$ a prime greater than $|a_0|$ , $k\nmid a_0$ , so $\gcd(a_0,k)=1$ ; by Step 1, $\gcd(P(k),k)=1$ , meaning every prime factor of $P(k)$ fails to divide $k$ . Since $P$ is non-constant, $|P(k)|\to\infty$ over such primes $k$ , so $P(k)$ always has a prime factor; because a non-constant integer polynomial cannot take values built only from a single fixed finite set of primes infinitely often, infinitely many distinct primes $p$ arise this way, each paired with some prime $k>|a_0|$ satisfying $p\mid P(k)$ , $p\nmid k$ .