Collatz problem

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

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 $c_0,c_1,c_2,\mathrm{\dots }$ 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