Pentation

The concept of "pentation" was named by English mathematician Reuben Goodstein in 1947, when he came up with the naming scheme for hyperoperations.

, depending on one's choice of notation.

For example, 2 pentated to the 2 is 2 tetrated to the 2, or 2 raised to the power of 2, which is

Based on this definition, pentation is only defined when a and b are both positive integers.

Pentation is the next hyperoperation (infinite sequence of arithmetic operations, based on the previous one each time) after tetration and before hexation.

It is defined as iterated (repeated) tetration (assuming right-associativity).

This is similar to as tetration is iterated right-associative exponentiation.

[1] It is a binary operation defined with two numbers a and b, where a is tetrated to itself b − 1 times.

The type of hyperoperation is typically denoted by a number in brackets, [].

For instance, using hyperoperation notation for pentation and tetration,

The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration.

It is part of his general naming scheme for hyperoperations.

[2] There is little consensus on the notation for pentation; as such, there are many different ways to write the operation.

However, some are more used than others, and some have clear advantages or disadvantages compared to others.

The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if

is defined by the Ackermann recurrence

[7] As tetration, its base operation, has not been extended to non-integer heights, pentation

is currently only defined for integer values of a and b where a > 0 and b ≥ −2, and a few other integer values which may be uniquely defined.

As with all hyperoperations of order 3 (exponentiation) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain: Additionally, we can also introduce the following defining relations: Other than the trivial cases shown above, pentation generates extremely large numbers very quickly.

As a result, there are only a few non-trivial cases that produce numbers that can be written in conventional notation, which are all listed below.

Some of these numbers are written in power tower notation due to their extreme size.