N8

Part of 2021 IMO Shortlist

Problems(1)

Iterated Integer Polynomial modulo n

Source: 2021 ISL N8

Find all positive integers

n

for which there exists a polynomial

P(x) \in \mathbb{Z}[x]

such that for every positive integer

m\geq 1

P^m(1), \ldots, P^m(n)

\lceil n/2^m\rceil

distinct remainders when divided by

n

P^m

P

m

times.)
Proposed by Carl Schildkraut, USA

algebrapolynomialceiling functionabstract algebra