Bowers' operators

Let m { p } n = H p ( m , n ) , the hyperoperation. That is

m { 1 } n = m + n

m { p } 1 = m  if  p ≥ 2

m { p } n = m { p − 1 } ( m { p } ( n − 1 ) )  if  n ≥ 2  and  p ≥ 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 ) )  if  n ≥ 2  and  p ≥ 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 ( Z + ) 3 , and the codomain of the operator is 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 ≥ 1 , n ≥ 2 , p ≥ 2 , and q ≥ 1 . The domain of ( m , n , p , q ) is ( Z + ) 4 , and the codomain of the operator is Z + .

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