Collatz problem

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

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

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.

Title	Collatz problem
Canonical name	CollatzProblem
Date of creation	2013-03-22 11:42:43
Last modified on	2013-03-22 11:42:43
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	32
Author	akrowne (2)
Entry type	Conjecture
Classification	msc 11B37
Synonym	Ulam’s Problem
Synonym	1-4-2 Problem
Synonym	Syracuse problem
Synonym	Thwaites conjecture
Synonym	Kakutani’s problem
Synonym	3n+1 problem