In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, means tetrating 2 to itself 2 times, or . This can then be reduced to
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 with the initial conditions and , then .
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 ≥ −1, 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 define:
Other than the trivial cases shown above, pentation generates extremely large numbers very quickly such that there are only a few non-trivial cases that produce numbers that can be written in conventional notation, as illustrated below: