How do you call / is there any specific name for the following:



  • 2^^2 = 2^2
  • 3^^3 = 3^3^3
  • 4^^4 = 4^4^4^4
  • ...
  • $\begingroup$ superpowers, Knuth notation. $\endgroup$ – Wuestenfux Dec 31 '18 at 17:31
  • 2
    $\begingroup$ Look up "tetration:. en.wikipedia.org/wiki/Tetration $\endgroup$ – Ethan Bolker Dec 31 '18 at 17:32
  • $\begingroup$ Power tower or tetration springs to mind, also denoted $$^na,$$ where $a$ is raised to itself $n-1$ times. $\endgroup$ – Antinous Dec 31 '18 at 17:33

This is known as tetration, where $^na$ is $a^{a^{a^{...}}}$ with $n$ copies of $a$ in the exponentiation. You might also seen it written as $a \uparrow \uparrow n$.

To learn more about these kinds of compound arithmetic operators, I recommend reading about Knuth's up-arrow notation.

  • $\begingroup$ Now that is one rapidly growing function! $\endgroup$ – Mike Jan 1 at 3:35
  • 1
    $\begingroup$ @Mike :P More like some of the slowest of fast growing functions $\endgroup$ – Simply Beautiful Art Feb 10 at 23:47

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