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The Collatz or the $3n+1$ conjecture is open.

  1. Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$ conjecture?

  2. What about the case of numbers of form $3^x$ (simplest non-trivial non-polynomial) where $x\in\mathbb Z$. Do these satisfy Collatz conjecture?

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    $\begingroup$ I suppose you're not interested in such trivial answers as $3\times2^n$, $5\times2^n$, and suchlike (but I feel compelled to mention them, just in case). $\endgroup$ Dec 22, 2021 at 23:04
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    $\begingroup$ No I am not interested. $\endgroup$
    – Turbo
    Dec 23, 2021 at 2:32
  • $\begingroup$ I think question 1 might be more natural without the absolute value. Or restricted to polynomials $\mathbb{N}\rightarrow \mathbb{N}$. $\endgroup$ Dec 24, 2021 at 8:48
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    $\begingroup$ Related question from OP: mathoverflow.net/questions/412416/… $\endgroup$ Dec 24, 2021 at 19:59
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    $\begingroup$ The Collatz conjecture is known to be true for specific numbers. 1,2, and 3 are examples. I know that is not what is intended, I am just suggesting a more specific title might be appropriate. $\endgroup$ Jan 13, 2022 at 4:23

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