# 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