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 3 of 5: Two periods of the sequence mod p

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P(k)\mid a_{i+k}-a_i,\qquad P(k+p)\equiv P(k)\equiv0\!\!\pmod p\implies p\mid a_{i+k+p}-a_i

Detailed analysis

For such a pair $(p,k)$ , $P(k)\mid a_{i+k}-a_i$ for every $i$ , so $p\mid a_{i+k}-a_i$ . Since $(k+p)-k=p$ , integer-coefficient polynomials satisfy $P(k+p)\equiv P(k)\equiv0\pmod p$ , so $p\mid P(k+p)$ , which gives $p\mid a_{i+k+p}-a_i$ for every $i$ as well.