Pentation

Pentation is the operation of repeated tetration, just as tetration is the operation of repeated exponentation and is a hyperoperation. It is non-commutative, and therefore has two inverse functions, which might be named the penta-root and the penta-logarithm (analogous to the two inverse functions for exponentiation: nth root function and logarithm). Pentation also bounds the elementary recursive functions.

The word "Pentation" was coined by Reuben Goodstein from the roots penta- (five) and iteration). It is part of his general naming scheme for hyperoperations.

Pentation can be written in Knuth's up-arrow notation as $a \uparrow \uparrow \uparrow b$ or $a \uparrow^{3}b$ .

1 Extension
- 1.1 Extension to zero and negative numbers
2 Selected values
3 References

Extension

It is not known how to extend pentation to complex or non-integer values.

Extension to zero and negative numbers

Using super-logarithm‎s, $a \uparrow^{3}b$ can be done when b is negative or 0 for some small integers. For all positive integer values of a negative pentation is as follows:

$a \uparrow^{3}0 = \operatorname{slog}_a a = 1, \,$
$a \uparrow^{3}-1 = \operatorname{slog}_a 1 = 0, \,$
$a \uparrow^{3}-2 = \operatorname{slog}_a 0 = -1, \,$

Any other case of this type of pentation produces an undefined result, since integer tetration does not take on the value -1.

It can also be done when a is negative, but this is only the case when a is equal to -1. For all positive integer values of b, the three possible answers that you can get for $-1 \uparrow^{3}b$ are shown below:

$-1 \uparrow^{3}b = {^{1}{-1}} = -1, \,$ if b is congruent to 1 modulo 3.
$-1 \uparrow^{3}b = {^{-1}{-1}} = 0, \,$ if b is congruent to 2 modulo 3.
$-1 \uparrow^{3}b = {^{0}{-1}} = 1, \,$ if b is congruent to 0 modulo 3.

If $\,\!a<-1$ , $a \uparrow^{3}2 = a \uparrow \uparrow a$ is undefined because tetration is only defined for b greater than -2, while if a is zero, we obtain the presumably-indeterminate form $0 \uparrow^{3}2 = {^{0}{0}}$ .

Selected values

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:

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

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:

$2 \uparrow^{3}2 = {^{2}2} = 4$
$2 \uparrow^{3}3 = {^{^{2}2}2} = ^{4}2 = 65,536$
$2 \uparrow^{3}4 = {^{^{^{2}2}2}2} = ^{65,536}2 = 2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}} \mbox{ (a power tower of height 65,536) } \approx \exp_{10}^{65,533}(4.29508)$ (shown here in iterated exponential notation as it's far too large to be written in conventional notation. Note $\exp_{10}(n) = 10^n$ )
$3 \uparrow^{3}2 = {^{3}3} = 7,625,597,484,987$
$3 \uparrow^{3}3 = {^{^{3}3}3} = {^{7,625,597,484,987}3} = 3^{3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}} \mbox{ (a power tower of height 7,625,597,484,987) } \approx \exp_{10}^{7,625,597,484,986}(1.09902)$
$4 \uparrow^{3}2 = {^{4}4} = 4^{4^{4^4}} = 4^{4^{256}} \approx \exp_{10}^3(2.19)$ (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

Tetration Forum by Jay D. Fox

Pentation

Contents

Extension

Extension to zero and negative numbers

Selected values

References