Bowers' operators

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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]


  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.CS1 maint: location (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>