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 5 of 5: Conclude the sequence is constant

Previous step Next step

a_{i+1}-a_i\text{ divisible by infinitely many primes}\implies a_{i+1}=a_i

Detailed analysis

Fix any $i$ . Step 2 supplies infinitely many primes $p$ , each yielding via Step 4 that $p\mid a_{i+1}-a_i$ . Since the fixed integer $a_{i+1}-a_i$ has infinitely many prime divisors, it must equal $0$ ; hence $a_{i+1}=a_i$ for every $i$ , and the sequence $a_1,a_2,a_3,\ldots$ is constant.