Pentation

In mathematics, pentation is the operation of repeated tetration, just as tetration is the operation of repeated exponentiation.

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.

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. Alternatively, in Conway chained arrow notation, $a\uparrow \uparrow \uparrow b=a\rightarrow b\rightarrow 3$. Another proposed notation is ${_{b}a}$, though this is not extensible to higher hyperoperations.

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 $a\uparrow \uparrow \uparrow b=A(4,b)$.

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 10¹⁰²¹⁸⁴ digits)