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 $a[5]b$, or using Knuth's up-arrow notation as $a \uparrow \uparrow \uparrow b$ or $a \uparrow^{3}b$. In this notation, $a\uparrow b$ represents the exponentiation function $a^b$, which may be interpreted as the result of repeatedly applying the function $x\mapsto ax$, for $b$ repetitions, starting from the number 1. Analogously, $a\uparrow\uparrow b$, tetration, represents the value obtained by repeatedly applying the function $x\mapsto a\uparrow x$, for $b$ repetitions, starting from the number 1. And the pentation $a \uparrow \uparrow \uparrow b$ represents the value obtained by repeatedly applying the function $x\mapsto a\uparrow\uparrow x$, for $b$ repetitions, starting from the number 1.^[3][4] Alternatively, in Conway chained arrow notation, $a\uparrow\uparrow\uparrow b = a\rightarrow b\rightarrow 3$.^[5] Another proposed notation is ${_{b}a}$, 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 $A(n,m)$ is defined by the Ackermann recurrence $A(m-1,A(m,n-1))$ with the initial conditions $A(1,n)=an$ and $A(m,1)=a$, then $a\uparrow\uparrow\uparrow b=A(4,b)$.^[7]

As its base operation (tetration) has not been extended to non-integer heights, pentation $a \uparrow^{3}b$ is currently only defined for integer values of a and b where a > 0 and b ≥ 0, 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:

• $1 \uparrow^{3}b = 1$
• $a \uparrow^{3}1 = a$

Additionally, we can also define:

• $a \uparrow^{3}0 = 1$
• $a \uparrow^{3}-1 = 0$

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:

• $2 \uparrow^{3}2 = {^{2}2} = 2^2 = 4$
• $2 \uparrow^{3}3 = {^{^{2}2}2} = {^{4}2} = 2^{2^{2^2}} = 2^{2^4} = 2^{16} = 65,536$
• $2 \uparrow^{3}4 = {^{^{^{2}2}2}2} = {^{65,536}2} = 2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}} \mbox{ (a power tower of height 65,536) } \approx \exp_{10}^{65,533}(4.29508)$ (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note $\exp_{10}(n) = 10^n$)
• $3 \uparrow^{3}2 = {^{3}3} = 3^{3^3} = 3^{27} = 7,625,597,484,987$
• $3 \uparrow^{3}3 = {^{^{3}3}3} = {^{7,625,597,484,987}3} = 3^{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}} \mbox{ (a power tower of height 7,625,597,484,987) } \approx \exp_{10}^{7,625,597,484,986}(1.09902)$
• $4 \uparrow^{3}2 = {^{4}4} = 4^{4^{4^4}} = 4^{4^{256}} \approx \exp_{10}^3(2.19)$ (a number with over 10¹⁵³ digits)
• $5 \uparrow^{3}2 = {^{5}5} = 5^{5^{5^{5^5}}} = 5^{5^{5^{3125}}} \approx \exp_{10}^4(3.33928)$ (a number with more than 10¹⁰²¹⁸⁴ digits)

References

1. Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic", Communications of the ACM 5 (6): 344, doi:10.1145/367766.368160.
2. Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic 12: 123–129, MR 0022537.
3. 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.
4. 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.
5. Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
6. http://www.tetration.org/Tetration/index.html
7. 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.