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

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

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

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

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

Tetrentrical operators

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

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

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

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

Thus, we have

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

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

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

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

⋮

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

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

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

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

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

⋮

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

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

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

Bowers continues with names for higher operations:

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

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

⋮

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

This means:

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

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

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

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

⋮

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

Bowers continues to provide names for the functions:

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

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

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

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

⋮

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

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

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

⋮

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

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

Pentetrical operators and beyond

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

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

Bowers does not provide operator names for

$\{a,b,1,1,2\}$

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

$3\{\{1\}\}64$

and

$3\{\{1\}\}65$

.