Bowers' operators

Let $m\{p\}n = H_p(m,n)$, the hyperoperation. That is

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

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

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

Invented by Jonathan Bowers, the first operator is $\{\{1\}\}$ and it's defined:

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

The number inside the brackets can change. If it's two

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

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

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

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

⋮

Thus, we have

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

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

$m\{\{2\}\}3 = m\{m\{m\}m\}m$

$m\{\{2\}\}4 = m\{m\{m\{m\}m\}m\}m$

⋮

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

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

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

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

Another function $\{m, n, p, q\}$ means $m\{\{...\{\{p\}\}...\}\}n$, 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 $(\mathbb{Z}^+)^4$, and the codomain of the operator is $\mathbb{Z}^+$.

Numbers like TREE(3) are unattainable with Bowers' operators, but Graham's number lies between $3\{\{2\}\}63$ and $3\{\{2\}\}64$.^[1]

References

1. Elwes, Richard (2010). Mathematics 1001: Absolutely Everything That Matters in Mathematics in 1001 Bite-Sized Explanations. Buffalo, New York 14205, United States: Firefly Books Inc. pp. 41–42. ISBN 978-1-55407-719-9.