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 ≥ −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:

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

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 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, PMID 17797067, doi:10.1126/science.194.4271.1235 .
↑ Blakley, G. R.; Borosh, I. (1979), "Knuth's iterated powers", Advances in Mathematics, 34 (2): 109–136, MR 549780, doi:10.1016/0001-8708(79)90052-5 .
↑ 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, MR 1368037, doi:10.1016/0893-9659(95)00084-4 .

Large numbers

Examples in numerical order

Expression methods

Notations	Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation
Operators	Hyperoperation Tetration Pentation Ackermann function Bowers' operators

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