# Bowers's operators

Bowers's operators was created by Jonathan Bowers.[1][2] It was created to help represent very large numbers, and was first published to the web in 2002.

## Definition

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

$\displaystyle{ m\{1\}n = m + n }$

$\displaystyle{ m\{p\}1 = m \text{ if } p \ge 2 }$

$\displaystyle{ m\{p\}n = m\{p-1\}(m\{p\}(n-1)) \text{ if } n \ge 2 \text{ and } p \ge 2 }$

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

### Tetrentrical operators

The first operator is $\displaystyle{ \{\{1\}\} }$ and it is defined:

$\displaystyle{ m\{\{1\}\}1 = m }$

$\displaystyle{ m\{\{1\}\}n = m\{m\{\{1\}\}n-1\}m }$

Bowers calls the function $\displaystyle{ m\{\{1\}\}n }$ "m expanded to n".

Thus, we have

$\displaystyle{ m\{\{1\}\}1 = m }$

$\displaystyle{ m\{\{1\}\}2 = m\{m\}m }$

$\displaystyle{ m\{\{1\}\}3 = m\{m\{m\}m\}m }$

$\displaystyle{ m\{\{1\}\}4 = m\{m\{m\{m\}m\}m\}m }$

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

$\displaystyle{ m\{\{2\}\}1 = m }$

$\displaystyle{ m\{\{2\}\}2 = m\{\{1\}\}(m\{\{2\}\}1) }$

$\displaystyle{ m\{\{2\}\}3 = m\{\{1\}\}(m\{\{2\}\}2) }$

$\displaystyle{ m\{\{2\}\}4 = m\{\{1\}\}(m\{\{2\}\}3) }$

Bowers calls the function $\displaystyle{ m\{\{2\}\}n }$ "m multiexpanded to n".

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

$\displaystyle{ m\{\{p\}\}n = m\{\{p-1\}\}(m\{\{p\}\}(n-1)) \text{ if } n \ge 2 \text{ and } p \ge 2 }$

Bowers continues with names for higher operations:

$\displaystyle{ m\{\{3\}\}n }$ is "m powerexpanded to n"

$\displaystyle{ m\{\{4\}\}n }$ is "m expandotetrated to n"

The next level of operators is $\displaystyle{ \{\{\{*\}\}\} }$, it to $\displaystyle{ \{\{*\}\} }$ behaves like $\displaystyle{ \{\{*\}\} }$ is to $\displaystyle{ \{*\} }$.

This means:

$\displaystyle{ m\{\{\{1\}\}\}1 = m }$

$\displaystyle{ m\{\{\{1\}\}\}2 = m\{\{m\}\}m }$

$\displaystyle{ m\{\{\{1\}\}\}3 = m\{\{m\{\{m\}\}m\}\}m }$

$\displaystyle{ m\{\{\{1\}\}\}4 = m\{\{m\{\{m\{\{m\}\}m\}\}m\}\}m }$

$\displaystyle{ m\{\{\{2\}\}\}n }$ and beyond will work similarly.

Bowers continues to provide names for the functions:

$\displaystyle{ m\{\{\{1\}\}\}n }$ is "m exploded to n"

$\displaystyle{ m\{\{\{2\}\}\}n }$ is "m multiexploded to n"

$\displaystyle{ m\{\{\{3\}\}\}n }$ is "m powerexploded to n"

$\displaystyle{ m\{\{\{4\}\}\}n }$ is "m explodotetrated to n"

$\displaystyle{ \{\{\{\{*\}\}\}\} }$ and beyond will follow similar recursion. Bowers continues with:

$\displaystyle{ m\{\{\{\{1\}\}\}\}n }$ is "m detonated to n"

$\displaystyle{ m\{\{\{\{\{1\}\}\}\}\}n }$ is "m pentonated to n"

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

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

### Pentetrical operators and beyond

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

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

Bowers does not provide operator names for $\displaystyle{ \{a,b,1,1,2\} }$ 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 $\displaystyle{ 3\{\{1\}\}64 }$ and $\displaystyle{ 3\{\{1\}\}65 }$.[3]