# Collatz problem

We define the function $f:\mathbb{N}\u27f6\mathbb{N}$ (where $\mathbb{N}$ excludes zero) such that

$$f(a)=\{\begin{array}{cc}\hfill 3a+1& \text{if}a\text{is odd}\hfill \\ \hfill a/2& \text{if}a\text{is even.}\hfill \end{array}$$ |

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 |