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Iterated Integer Polynomial modulo n

Source: 2021 ISL N8

July 12, 2022
algebrapolynomialceiling functionabstract algebra

Problem Statement

Find all positive integers nn for which there exists a polynomial P(x)∈Z[x]P(x) \in \mathbb{Z}[x] such that for every positive integer m≥1m\geq 1, the numbers Pm(1),…,Pm(n)P^m(1), \ldots, P^m(n) leave exactly ⌈n/2m⌉\lceil n/2^m\rceil distinct remainders when divided by nn. (Here, PmP^m means PP applied mm times.)
Proposed by Carl Schildkraut, USA
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