# Bowers's operators

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

## Functionality

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

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

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

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

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 number inside the brackets can change. If it's 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:

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

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 ${\displaystyle m\geq 1}$, ${\displaystyle n\geq 2}$, ${\displaystyle p\geq 2}$, and ${\displaystyle q\geq 1}$. 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]