Bowers's operators

Bowers's operators was created by Jonathan Bowers. It was created to help represent very large numbers, and was first published to the web in 2002.

Definition

Let [math]\displaystyle{ m\{p\}n = H_p(m,n) }[/math], the hyperoperation (see Square bracket notation, this [math]\displaystyle{ m[p]n }[/math] is the same as [math]\displaystyle{ m\{p\}n }[/math], they are just different notations of hyperoperation). That is

[math]\displaystyle{ m\{1\}n = m + n }[/math]

[math]\displaystyle{ m\{p\}1 = m \text{ if } p \ge 2 }[/math]

[math]\displaystyle{ m\{p\}n = m\{p-1\}(m\{p\}(n-1)) \text{ if } n \ge 2 \text{ and } p \ge 2 }[/math]

The function [math]\displaystyle{ \{m, n, p\} }[/math] means [math]\displaystyle{ m\{p\}n }[/math], i.e. [math]\displaystyle{ \{m, n, p\} }[/math] is equal to [math]\displaystyle{ H_p(m,n) }[/math] for every [math]\displaystyle{ (m, n, p) }[/math] ∈ [math]\displaystyle{ (\mathbb{Z}^+)^3 }[/math].

Tetrentrical operators

The first operator is [math]\displaystyle{ \{\{1\}\} }[/math] and it is defined:

[math]\displaystyle{ m\{\{1\}\}1 = m }[/math]

[math]\displaystyle{ m\{\{1\}\}n = m\{m\{\{1\}\}n-1\}m }[/math]

Bowers calls the function [math]\displaystyle{ m\{\{1\}\}n }[/math] "m expanded to n".

Thus, we have

[math]\displaystyle{ m\{\{1\}\}1 = m }[/math]

[math]\displaystyle{ m\{\{1\}\}2 = m\{m\}m }[/math]

[math]\displaystyle{ m\{\{1\}\}3 = m\{m\{m\}m\}m }[/math]

[math]\displaystyle{ m\{\{1\}\}4 = m\{m\{m\{m\}m\}m\}m }[/math]

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The argument inside the brackets can be increased. If the argument is increased to two:

[math]\displaystyle{ m\{\{2\}\}1 = m }[/math]

[math]\displaystyle{ m\{\{2\}\}2 = m\{\{1\}\}(m\{\{2\}\}1) }[/math]

[math]\displaystyle{ m\{\{2\}\}3 = m\{\{1\}\}(m\{\{2\}\}2) }[/math]

[math]\displaystyle{ m\{\{2\}\}4 = m\{\{1\}\}(m\{\{2\}\}3) }[/math]

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Bowers calls the function [math]\displaystyle{ m\{\{2\}\}n }[/math] "m multiexpanded to n".

Operators beyond [math]\displaystyle{ \{\{2\}\} }[/math] can also be made, the rule of it is the same as hyperoperation:

[math]\displaystyle{ m\{\{p\}\}n = m\{\{p-1\}\}(m\{\{p\}\}(n-1)) \text{ if } n \ge 2 \text{ and } p \ge 2 }[/math]

Bowers continues with names for higher operations:

[math]\displaystyle{ m\{\{3\}\}n }[/math] is "m powerexpanded to n"

[math]\displaystyle{ m\{\{4\}\}n }[/math] is "m expandotetrated to n"

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The next level of operators is [math]\displaystyle{ \{\{\{*\}\}\} }[/math], it to [math]\displaystyle{ \{\{*\}\} }[/math] behaves like [math]\displaystyle{ \{\{*\}\} }[/math] is to [math]\displaystyle{ \{*\} }[/math].

This means:

[math]\displaystyle{ m\{\{\{1\}\}\}1 = m }[/math]

[math]\displaystyle{ m\{\{\{1\}\}\}2 = m\{\{m\}\}m }[/math]

[math]\displaystyle{ m\{\{\{1\}\}\}3 = m\{\{m\{\{m\}\}m\}\}m }[/math]

[math]\displaystyle{ m\{\{\{1\}\}\}4 = m\{\{m\{\{m\{\{m\}\}m\}\}m\}\}m }[/math]

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[math]\displaystyle{ m\{\{\{2\}\}\}n }[/math] and beyond will work similarly.

Bowers continues to provide names for the functions:

[math]\displaystyle{ m\{\{\{1\}\}\}n }[/math] is "m exploded to n"

[math]\displaystyle{ m\{\{\{2\}\}\}n }[/math] is "m multiexploded to n"

[math]\displaystyle{ m\{\{\{3\}\}\}n }[/math] is "m powerexploded to n"

[math]\displaystyle{ m\{\{\{4\}\}\}n }[/math] is "m explodotetrated to n"

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[math]\displaystyle{ \{\{\{\{*\}\}\}\} }[/math] and beyond will follow similar recursion. Bowers continues with:

[math]\displaystyle{ m\{\{\{\{1\}\}\}\}n }[/math] is "m detonated to n"

[math]\displaystyle{ m\{\{\{\{\{1\}\}\}\}\}n }[/math] is "m pentonated to n"

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For every fixed positive integer [math]\displaystyle{ q }[/math], there is an operator [math]\displaystyle{ m\{\{\ldots\{\{p\}\}\ldots\}\}n }[/math] with [math]\displaystyle{ q }[/math] sets of brackets. The domain of [math]\displaystyle{ (m, n, p) }[/math] is [math]\displaystyle{ (\mathbb{Z}^+)^3 }[/math], and the codomain of the operator is [math]\displaystyle{ \mathbb{Z}^+ }[/math].

Another function [math]\displaystyle{ \{m, n, p, q\} }[/math] means [math]\displaystyle{ m\{\{\ldots\{\{p\}\}\ldots\}\}n }[/math], where [math]\displaystyle{ q }[/math] is the number of sets of brackets. It satisfies that [math]\displaystyle{ \{m, n, p, q\} = \{m, \{m, n-1, p, q\},p-1, q\} }[/math] for all integers [math]\displaystyle{ m \ge 1 }[/math], [math]\displaystyle{ n \ge 2 }[/math], [math]\displaystyle{ p \ge 2 }[/math], and [math]\displaystyle{ q \ge 1 }[/math]. The domain of [math]\displaystyle{ (m, n, p, q) }[/math] is [math]\displaystyle{ (\mathbb{Z}^+)^4 }[/math], and the codomain of the operator is [math]\displaystyle{ \mathbb{Z}^+ }[/math].

Pentetrical operators and beyond

Bowers generalizes this towards 5+ entries with the following ruleset:

1. [math]\displaystyle{ \{\}=1 }[/math] if there are 0 entries (like Conway chained arrow notation, the value of the empty chain is 1),
2. [math]\displaystyle{ \{a\}=a,\{a,b\}=a^b }[/math] if there are only 1 or 2 entries,
3. [math]\displaystyle{ \{a,b,c,...,k,1\} = \{a,b,c,...,k\} }[/math] if the last entry is 1,
4. [math]\displaystyle{ \{a,1,c,d,...,k\} = a }[/math] if the second entry is 1,
5. [math]\displaystyle{ \{a,b,1,...,1,d,e,...,k\}=\{a,a,a,...,\{a,b-1,1,...,1,d,e,...,k\},d-1,e,...,k\} }[/math] if the 3rd entry is 1,
6. [math]\displaystyle{ \{a,b,c,d,...,k\}=\{a,\{a,b-1,c,d,...,k\},c-1,d,...,k\} }[/math] if none of the above rules apply.

Bowers does not provide operator names for [math]\displaystyle{ \{a,b,1,1,2\} }[/math] and beyond, but he describes a notation for up to 8 entries:

"a,b, and c are shown in numeric form, d is represented by brackets (as seen above), e is shown by [ ] like brackets, but rotated 90 degrees, where the brackets are above and below (uses e-1 bracket sets), f is shown by drawing f-1 Saturn like rings around it, g is shown by drawing g-1 X-wing brackets around it, while h is shown by sandwiching all this in between h-1 3-D versions of [ ] brackets (above and below) which look like square plates with short side walls facing inwards."

Numbers like TREE(3) are unattainable with Bowers's operators, but Graham's number lies between [math]\displaystyle{ 3\{\{1\}\}64 }[/math] and [math]\displaystyle{ 3\{\{1\}\}65 }[/math].