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'Collatz problem'
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| Title of object: |
Collatz problem |
| Canonical Name: |
CollatzProblem |
| Type: |
Conjecture |
| Created on: |
2001-08-19 10:20:17 |
| Modified on: |
2002-08-01 13:09:07 |
| Classification: |
msc:11B37 |
| Keywords: |
collatz, ulam |
| Synonyms: |
Collatz problem=Ulam's Problem
Collatz problem=1-4-2 Problem
Collatz problem=Syracuse problem
Collatz problem=Thwaites conjecture
Collatz problem=Kakutani's problem
Collatz problem=3n+1 problem |
Revision comment (for changes between this and next version):
| Changes for correction #2480 ('redefining f(x)'). (Clarifying the Naturals) |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
We define the function $f : \mathbb{N} \longrightarrow \mathbb{N}$ so that
$$ f(n+1) = \left\{
\begin{array}{rl}
3f(n)+1$ & f(n) \text{ is odd } \\
f(n)/2$ & f(n) \text{ is even.}
\end{array}
\right. $$
$f(0)$ is an arbitrary seed value.
It is conjectured that the sequence $f(0),f(1),f(2),\ldots$ will always end in $1,4,2$, which repeats infinitely. This has been verified by computer up to very large values of $f(0)$, but is unproven in general. It is also not known whether this problem is decideable.
This is sometimes called the ``hailstone sequence'' because, like a hailstone in a cloud, the values oscillate up and down. |
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