# Pentation

In mathematics, **pentation** is the operation of repeated tetration, just as tetration is the operation of repeated exponentiation.^{[1]}

## History

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]}

## Notation

Pentation can be written as a hyperoperation as , or using Knuth's up-arrow notation as or . In this notation, represents the exponentiation function , which may be interpreted as the result of repeatedly applying the function , for repetitions, starting from the number 1. Analogously, , tetration, represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. And the pentation represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1.^{[3]}^{[4]} Alternatively, in Conway chained arrow notation, .^{[5]} Another proposed notation is , though this is not extensible to higher hyperoperations.^{[6]}

## Examples

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 .^{[7]}

As its base operation (tetration) 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. Like all other 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:

- (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note )
- (a number with over 10
^{153}digits) - (a number with more than 10
^{102184}digits)

## References

- ↑ Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic",
*Communications of the ACM*,**5**(6): 344, doi:10.1145/367766.368160. - ↑ Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory",
*The Journal of Symbolic Logic*,**12**: 123–129, MR 0022537. - ↑ Knuth, D. E. (1976), "Mathematics and computer science: Coping with finiteness",
*Science*,**194**(4271): 1235–1242, doi:10.1126/science.194.4271.1235, PMID 17797067. - ↑ Blakley, G. R.; Borosh, I. (1979), "Knuth's iterated powers",
*Advances in Mathematics*,**34**(2): 109–136, doi:10.1016/0001-8708(79)90052-5, MR 549780. - ↑ Conway, John Horton; Guy, Richard (1996),
*The Book of Numbers*, Springer, p. 61, ISBN 9780387979939. - ↑ http://www.tetration.org/Tetration/index.html
- ↑ Nambiar, K. K. (1995), "Ackermann functions and transfinite ordinals",
*Applied Mathematics Letters*,**8**(6): 51–53, doi:10.1016/0893-9659(95)00084-4, MR 1368037.