Problem 3

For each integer

k\ge2

, determine all infinite sequences of positive integers

a_1,a_2,\ldots

for which there exists a polynomial

P

of the form

P(x)=x^k+c_{k-1}x^{k-1}+\cdots+c_1x+c_0

, where

c_0,c_1,\ldots,c_{k-1}

are non-negative integers, such that

P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k}

for every integer

n\ge1

Step 1 of 6: Arithmetic progressions work

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In plain words

If the terms increase by a fixed step d, the k terms right after a_n are simply a_n plus the first k multiples of d, so their product is a fixed polynomial evaluated at a_n.

a_n=a_1+(n-1)d\implies P(x)=\prod_{j=1}^k(x+jd)

Detailed analysis

If $a_n=a_1+(n-1)d$ for an integer $d\ge0$ , then $a_{n+1}a_{n+2}\cdots a_{n+k}=(a_n+d)(a_n+2d)\cdots(a_n+kd)$ , so $P(x)=(x+d)(x+2d)\cdots(x+kd)$ works, and its coefficients besides the leading one are non-negative since $d\ge0$ .