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 4 of 5: Combine the two periods

Previous step Next step

\gcd(k,k+p)=1\implies p\mid a_{i+1}-a_i\ \text{for every }i

Detailed analysis

By Step 3, the sequence $(a_i\bmod p)$ is periodic with periods $k$ and $k+p$ . Since $p\nmid k$ , $\gcd(k,k+p)=\gcd(k,p)=1$ , and a sequence periodic with two coprime periods is periodic with their gcd, which is $1$ ; that is, $p\mid a_{i+1}-a_i$ for every $i$ .