Bowers' operators

Let , the hyperoperation. That is

Invented by Jonathan Bowers, the first operator is and it is defined:

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

Thus, we have

Operators beyond can also be made, the rule of it is the same as hyperoperation:

The next level of operators is , it to behaves like is to .

For every fixed positive integer , there is an operator with sets of brackets. The domain of is , and the codomain of the operator is .

Another function means , where is the number of sets of brackets. It satisfies that for all integers , , , and . The domain of is , and the codomain of the operator is .

Numbers like TREE(3) are unattainable with Bowers' operators, but Graham's number lies between and .[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.
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