In mathematics, pentation is the next hyperoperation after tetration but 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 times. For instance, using hyperoperation notation for pentation and tetration, means tetrating 2 to itself 3 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.
- 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. 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.
- Another proposed notation is , though this is not extensible to higher hyperoperations.
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:
- (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note )
- (a number with over 10153 digits)
- (a number with more than 10102184 digits)
- 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 (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, 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 0549780.
- Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
- 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.