Knuth's up-arrow notation

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Knuth's up-arrow notation is a notation for expressing large numbers. However, there are some numbers so large that even this up-arrow notation is not enough. You have to use Extended arrow notation.

Definition

\(a \uparrow b=a^b\)

\(a \uparrow\uparrow b=\underbrace{a^{a^{.^{.^{.^{a}}}}}}_{\text{b a's}}\)

\(a \uparrow\uparrow\uparrow b=\underbrace{a \uparrow\uparrow a \uparrow\uparrow a\cdots\uparrow\uparrow a}_{\text{b a's}}\)

\(a \uparrow^{n+1} b=\underbrace{a \uparrow^n a \uparrow^n a\cdots a\uparrow^n a}_{\text{b a's}}\)

These operators are right-associative. Some of them are given names:

  • $\uparrow\uparrow$ is called "tetration"
  • $\uparrow\uparrow\uparrow$ is often called "pentation"
  • $\uparrow\uparrow\uparrow\uparrow$ is often called "hexation"

Examples

\(3\uparrow\uparrow\uparrow3=3\uparrow\uparrow3\uparrow\uparrow3=3\uparrow\uparrow3\uparrow3\uparrow3=3\uparrow\uparrow3\uparrow27=3\uparrow\uparrow7625597484987=\)a power tower that will reach all the way to the Sun