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This question is inspired by a riddle in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question. Still open:

1. In the setting of the original question, is it true that some composition power of $P$ takes integers to integers?
2. The original question with rationals.

This question is inspired by a riddle in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question. Still open:

1. In the setting of the original question, is it true that some composition power of $P$ takes integers to integers?
2. The original question with rationals.

This question is inspired by a riddle in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question. Still open:

1. If $P$ takes integers to integers and its orbit under zero contains infinitely many (different) integers, is it true that some composition power of it is integral?
2. The original question with rationals.

This question is inspired by a riddle in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question. Still open:

1. In the setting of the original question, is it true that some composition power of $P$ takes integers to integers?
2. The original question with rationals.

This question is inspired by a riddle in math.stackexchange.

Let $P(x)$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

This question is inspired by a riddle in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question. Still open:

1. If $P$ takes integers to integers and its orbit under zero contains infinitely many (different) integers, is it true that some composition power of it is integral?
2. The original question with rationals.