Problem 4

Let

\mathbb{Z}

denote the set of all integers. Find all polynomials

P(x)

with integer coefficients that satisfy the following property: for any infinite sequence

a_1,a_2,\ldots

of integers in which each integer in

\mathbb{Z}

appears exactly once, there exist indices

i<j

and an integer

k

such that

a_i+a_{i+1}+\cdots+a_j=P(k)

Step 3 of 5: That block sum is P(k)

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a_{e_p}+a_{e_p+1}+\cdots+a_{e_q}\equiv d\pmod c

Detailed analysis

Then $s_{e_q}-s_{e_p-1}=(s_{e_q-1}+a_{e_q})-s_{e_p-1}\equiv(s_{e_p-1}+d)-s_{e_p-1}=d\pmod c$ , and this difference is exactly the block sum $a_{e_p}+a_{e_p+1}+\cdots+a_{e_q}$ , a sum of at least two consecutive terms since $e_p<e_q$ . Being congruent to $d\pmod c$ , this sum equals $ck+d=P(k)$ for a suitable integer $k$ , so every linear $P$ with $c\ne0$ has the property.