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Collatz problem

We define the function $f : \mathbb{N} \longrightarrow \mathbb{N} $ (where $\mathbb{N}$ excludes zero) such that

$\begin{displaymath}f(a) = \left\{ \begin{array}{rl} 3a+1 & \text{ if } a \text{ ... ... } \ a/2 & \text{ if } a \text{ is even.} \end{array}\right. \end{displaymath}$

Then let the sequence $c_n$ be defined as $c_i = f(c_{i-1})$ , with $c_0$ an arbitrary natural seed value.

It is conjectured that the sequence $c_0, c_1, c_2, \ldots$ will always end in $1,4,2$ , repeating infinitely. This has been verified by computer up to very large values of $c_0$ , but is unproven in general. It is also not known whether this problem is decideable. This is generally called the Collatz problem.

The sequence $c_n$ is sometimes called the ``hailstone sequence''. This is because it behaves analogously to a hailstone in a cloud which falls by gravity and is tossed up again repeatedly. The sequence similarly ends in an eternal oscillation.

Collatz problem is owned by Aaron Krowne.

How to Cite This Entry

Aaron Krowne. "Collatz problem" (version 16). PlanetMath.org. Freely available at http://planetmath.org/CollatzProblem.html.

Classification

AMS MSC:

11B37 (Number theory :: Sequences and sets :: Recurrences)

Pending Errata and Addenda

None. [ View all 7 ]

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Discussion

Is following theorem known? by Makoto Kohno on 2007-06-22 08:47:38

Let $ S$ be the set of odd positive integers.

We define $R:S \longrightarrow \ S $ such that $R(n) = (3n+1)/2^r$, with $ r$ as large as possible.

Next we define $ M(n)=sup(n,R(n),R(Rn)),...}$ for every odd positive integer.

Then

THEOREM
Let $ K$ be a positive number.
Then there exist odd integer $ n$ such that $ M(n)/n > K$ .

[ reply | up ]

Re: Is following theorem known? by silverfish on 2007-06-22 09:52:27
- Re: Is following theorem known? by Makoto Kohno on 2007-06-22 10:17:55

Collatz Problem by Christopher Lee on 2001-10-06 01:22:48

This has been a conjecture that has stuck in the back of my mind for a very long time. Was reading some literature on it the other day, and it seems that someone has come up with a topological characterization. Basically, all the orbits of the Collatz function form a basis for a topology on the positive integers induced by this function. A corollary to a theorem states that the Collatz Conjecture is true if and only if the topology induced by the Collatz Function on the positive integers is connected. Another interaction of number theory, topology and dynamical systems...