**PAGE UNDER CONSTRUCTION**

This page displays known bounds for Rado's sigma function. See a previous version here.

## Credits

My machines wouldn't have existed without the following people:

- Many of these machines uses Deedlit's expandal machine and LittlePeng9's duplication machine.

- Page inspired by a result based on LittlePeng9's duplication machine.

- \(\Sigma(7)\) bound is a simple combination of Kropitz' and Michel's results.

- Michel and Marxen have published useful simulatations of machines.

I am grateful to these people for all their work. Without their work, a lot of my work wouldn't exist and I would probably not be so interested in the Busy Beaver problem. I would also like to thank the whole googology community for their support and general assistance.

## My results so far

Here are my results so far on this page:

- Proving the best known bounds for \(\Sigma(7,2)\) and \(\Sigma(n,2)\) for \(13 \leq n \leq 22\).
- Proving the best known bounds for \(\Sigma(6,3)\) and \(\Sigma(7,3)\).

## All known bounds

### Bounds for \(\Sigma(n,2)\) and \(S(n,2)\)

This section gives all bounds for \(\Sigma(n)\) known to me.

n |
Lower bound \(\Sigma(n)\) | Lower bound \(S(n)\) | Subpages |
---|---|---|---|

1 | \(1\) (exactly) | \(1\) (exactly) | (see details) |

2 | \(4\) (exactly) | \(6\) (exactly) | (see details) |

3 | \(6\) (exactly) | \(21\) (exactly) | (see details) |

4 | \(13\) (exactly) | \(107\) (exactly) | (see details) |

5 | \(4098\) | \(47176870\) | (see details) |

6 | \(3.5 \times 10^{18267}\) | \(7.4\times10^{36534}\) | (see details) |

7 | \(10^{10^{10^{10^{18705352}}}}\) | \(10^{10^{10^{10^{18705352}}}}\) | (see details) |

Bounds for \(\Sigma(9)\) through \(\Sigma(12)\) can be given using Milton W. Green's work. Let \(A=2161856886993815\).

n | Lower bound \(\Sigma(n)\) | Subpages |
---|---|---|

9 | \(\{3,31,2\}\) | (see details) |

10 | \(\{3,\{3,A\},2\}\) | (see details) |

11 | \(\{3,A,3\}\) | (see details) |

12 | \(\{3,\{3,\{3,A+1,2\},2\},3\}\) | (see details) |

Bounds for \(\Sigma(13)\) and further are given below.

n | Lower bound \(\Sigma(n)\) | Subpages |
---|---|---|

13 | \(\{3,3,2046\}\) | (see details) |

14 | \(\{3,3,1.7\times10^{18267}\}\) | (see details) |

15 | \(\{3,3,10^{10^{10^{10^{18705352}}}}\}\) | (see details) |

### Bounds for \(\Sigma(n,3)\) and \(S(n,3)\)

n | Lower bound \(\Sigma(n,3)\) | Lower bound \(S(n,3)\) |
---|---|---|

1 | \(1\) (exactly) | \(1\) (exactly) |

2 | \(9\) (exactly) | \(38\) (exactly) |

3 | \(374676383\) | \(119112334170342540\) |

4 | \(1.3 \times 10^{7036}\) | \(1.0 \times 10^{14072}\) |

Bounds for \(\Sigma(6,3)\) and further are given below.

n | Lower bound \(\Sigma(n,3)\) | Subpages |
---|---|---|

6 | \(\{2,8,6\}\) | (see details) |

7 | \(\{2,\{2,374676382,374676380\},374676380\}\) | (see details) |

## Subpages

- Oracle TMs (This page gives a definition for Oracle TMs, and 2 bounds)
- Analysis of LittlePeng9's work (This page gives bounds and comments on LittlePeng9's work)
- Progress on the 42 Hardly Non-Regular TMs (This page describes the progress on the 42 Hardly Non-Regular TMs, and thus the progress on proving \(\Sigma(5) = 4098\).)
- Multi-Headed TMs (This page gives a definition for Multi-Headed TMs, and 7 bounds)