Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.
The $n$th tetration of $a$, most often written $^na$, is defined as $n$ copies of $a$ combined by exponentiation in the style of power-towers. The evaluation proceeds right-to-left as is the norm for nested exponentials:
$$^na=\underbrace{a^{a^{\ \dots\ ^{a}}}}_n$$
This classical definition works whenever $a$ (the base) is an integer or positive real number and $n$ (the height) is a non-negative integer. Alternative notations include $a\uparrow\uparrow n$ (Knuth's), $a\to n\to2$ (Conway's) and the text notation a^^n.
The alternative name hyper-4 for tetration reflects its place as the fourth hyperoperation after addition, multiplication and exponentiation. There are three main types of questions relating to tetration:
- If the base is a natural number, the result of tetration will be a very large natural number and number-theoretical questions like "What is $^na\bmod N$?" are relevant. For example, computing the last digits of Graham's number involves computing the last digits of $^n3$ for sufficiently large $n$.
- How can tetration be extended beyond the classical definition? Complex bases can be easily accommodated, while the extension to infinite heights ($^\infty z$) features a connection to the Lambert W function: $$^\infty z=\frac{W(-\ln z)}{-\ln z}$$ In contrast, there are several proposed extensions of tetration to real or complex heights, but none have been widely accepted.
- How can tetration be reversed? Just as the two inverses of exponentiation are roots and logarithms, the two inverses of tetration are superroots and superlogarithms. As with tetration's extensions, there are many open questions relating to these two inverses.
