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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

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