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I was asked this question that, for any $x_1 \in \mathbb{N}$, define the sequence as

$$x_{n+1}=\left\{ \begin{array}{l l} x_n/2 & \quad \text{if } x \text{ is even} \\ 3 x_n+1 & \quad \text{if } x \text{ is odd} \end{array} \right.$$

Will $x_n$ always shrink to 1?

ps. My knowledge on Number theory is really close to $0$. I'm not sure if this is an elementary or advanced question.

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    $\begingroup$ This is the Collatz conjecture, and is very open. $\endgroup$ – user61527 Oct 7 '13 at 1:10
  • $\begingroup$ This would be a very advanced question in terms of how difficult it is to solve! Consider that Erdos (a great mathematician) famously stated: "mathematics is not yet ready for such problems"! For more, see mathworld.wolfram.com/CollatzProblem.html. $\endgroup$ – Benjamin Dickman Oct 7 '13 at 1:17
  • $\begingroup$ @T.Bongers thanks, i got it! $\endgroup$ – athos Oct 7 '13 at 1:22
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just to close my question. it's actually answered in the comments.

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