Pentation
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
![{\displaystyle 2[5]3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30c3c4f3a4215af3d0aba5d1be34d345eb03d108)
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80ca481db73e672c49db94005ca4638ee07e9a35)
=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddaf74f44639cff497f1a63566693ca571a198e7)
Etymology
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
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.
![{\displaystyle a[5]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d782af460e582816fb4e49d3907b621dd297d4d3)
![{\displaystyle a[3]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd946603b3dd3c072053d13fbe120d44f70ac8ab)
![{\displaystyle x\mapsto a[2]x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de6cda8581e622a5228d53c7ad2e3eb16ca119b)
![{\displaystyle a[4]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfddc8a9d38692fe9ac6ecee1527c90dd2ecccca)
![{\displaystyle x\mapsto a[3]x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81ffd1906771db6725baa931ab37956655a65e38)
![{\displaystyle x\mapsto a[4]x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0a37e85c7d20e456e6ef5db50d1c3b6d2f1915)
- 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.
![{\displaystyle a[5]b=a\rightarrow b\rightarrow 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63a65e4bf6c07338977b0fedeca2a2b702f4306)
- 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]
![{\displaystyle a[5]b=A(4,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de36fd296e5556dc05d3da63d988f0e3aba29d19)
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:
![{\displaystyle 1[5]b=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efa921244f5a19ac238e1e5facbce803f7bd536a)
![{\displaystyle a[5]1=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b09635a9e2ad78029099d065c824340b94ee2c)
Additionally, we can also define:
![{\displaystyle a[5]0=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad5cdf53a1e0c4bd1ee33b75c0375f3c2be4bbe)
=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c38979a979502245eb33bd881f3b464e45adfea0)
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)
![{\displaystyle 2[5]2=2[4]2=2^{2}=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f544aee58a80be2fa1dcdae14cddab48f8b189d9)
![{\displaystyle 2[5]3=2[4](2[4]2)=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da751a506265725b235de9808832aace07e154bc)
![{\displaystyle 2[5]4=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed3ef7ab241291154b648a4aa994cff4e16f759)
![{\displaystyle 3[5]2=3[4]3=3^{3^{3}}=3^{27}=7,625,597,484,987}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a455992a369ea94a39c61963513f4d19b3c7800)
![{\displaystyle 3[5]3=3[4](3[4]3)=3[4]7,625,597,484,987={\underset {{\text{3 is repeated 3[4]3 times}}=3^{3^{3}}}{3^{3^{.^{.^{.^{3}}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d64489899f025a3a302bce87a7639cfac6752bf3)
![{\displaystyle 4[5]2=4[4]4=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cb9353578126aa7a2e5dbbb93f5587a9f80373a)
![{\displaystyle 5[5]2=5[4]5=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx \exp _{10}^{4}(3.33928)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/428de46090279b2771bd9e491bfb0d4886183d53)
See also
References
- ^ 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.