Bowers' operators

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

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

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

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

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

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

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 $m\geq 1$ , $n\geq 2$ , $p\geq 2$ , and $q\geq 1$ . 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

↑ 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.

Large numbers

Examples in numerical order

Expression methods

Notations	Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation

Operators	Hyperoperation Tetration Pentation Ackermann function Bowers' operators

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