KNOWPIA
WELCOME TO KNOWPIA

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.^{[1]} 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

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.

- Pentation can be written as a hyperoperation as . In this format, 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]}This will be the notation used in the rest of the article.

- In Knuth's up-arrow notation, is represented as or . In this notation, represents the exponentiation function and represents tetration. The operation can be easily adapted for hexation by adding another arrow.

- In Conway chained arrow notation, .
^{[5]}

- Another proposed notation is , though this is not extensible to higher hyperoperations.
^{[6]}

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 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:

- (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)

**^**Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic",*Communications of the ACM*,**5**(6): 344, doi:10.1145/367766.368160, S2CID 581764.**^**Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory",*The Journal of Symbolic Logic*,**12**(4): 123–129, doi:10.2307/2266486, JSTOR 2266486, MR 0022537.**^**Knuth, D. E. (1976), "Mathematics and computer science: Coping with finiteness",*Science*,**194**(4271): 1235–1242, Bibcode:1976Sci...194.1235K, doi:10.1126/science.194.4271.1235, PMID 17797067, S2CID 1690489.**^**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 0549780.**^**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.