Fourth Hyperoperation is Tetration Operation
Theorem
The $4$th hyperoperation is the tetration operation restricted to the positive integers:
- $\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$
where $\uparrow \uparrow$ denotes tetration:
- $x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$
In this context, $x \uparrow y$ is the Knuth notation for powers:
- $x \uparrow y := x^y$
Proof
By definition of the hyperoperation sequence:
- $\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$
Thus the $4$th hyperoperation is defined as:
- $\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = \begin{cases} 1 & : y = 0 \\ H_3 \left({x, H_4 \left({x, y - 1}\right)}\right) & : y > 0 \end{cases}$
From Third Hyperoperation is Integer Power Operation:
- $(1): \quad \forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = \begin{cases} 1 & : y = 0 \\ x \uparrow H_3 \left({x, y - 1}\right) & : y > 0 \end{cases}$
The proof proceeds by induction.
For all $y \in \Z_{\ge 0}$, let $P \left({y}\right)$ be the proposition:
- $\forall x \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$
Basis for the Induction
$P \left({0}\right)$ is the case:
\(\ds H_4 \left({x, 0}\right)\) | \(=\) | \(\ds 1\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \uparrow \uparrow 0\) | Definition of Tetration |
Thus $P \left({0}\right)$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k + 1}\right)$ is true.
So this is the induction hypothesis:
- $\forall x \in \Z_{\ge 0}: H_4 \left({x, k}\right) = x \uparrow \uparrow k$
from which it is to be shown that:
- $\forall x \in \Z_{\ge 0}: H_4 \left({x, k + 1}\right) = x \uparrow \uparrow \left({k + 1}\right)$
Induction Step
This is the induction step:
\(\ds H_4 \left({x, k + 1}\right)\) | \(=\) | \(\ds x \uparrow H_4 \left({x, \left({k + 1}\right) - 1}\right)\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \uparrow H_4 \left({x, k}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \uparrow \left({x \uparrow \uparrow k}\right)\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds x \uparrow \uparrow \left({k + 1}\right)\) | Definition of Tetration |
So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x \uparrow \uparrow y$
$\blacksquare$