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