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$n*n$ is a square number. Is there a corresponding descriptive term for $n^n$?

Auto-power? 2nd-order tetration?

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up vote 2 down vote accepted

There was the name "Wexzal" introduced. Fantini/Kloepfer discussed it in that way:

This book is about the solution to and properties of the Coupled Exponent equation (y=x^x). The solution to this equation is called the "Coupled Root function".

(but I did not read that name elsewhere, so it seems that name didn't make it through the world)

"Wexzal", Jay A. Fantini Gilbert C. Kloepfer (~1999)

online available, maybe you must employ the wayback-machine.

In he tetration-forum it became fairly common to call the iteration of iterated exponentiation/powertower "height", so we would call it "powertower of height 2" or some easier-to-speak variants.
ANother hint, which you possibly have not come across yet: you could also look at R. Munafo's site who invented some expressions for iterated exponentiation and resulting "really big" numbers.

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That source has: In 1981, the term "Wexzal" was defined to mean "Coupled Root of 10^x". The term that source uses for $x^x$ is "coupled exponent". It says alternate names are "Self-exponential" and "Second-order Towering exponent". pdf – xan Jan 20 '13 at 2:09

I've most often heard simply "$n$ to the $n$". The term "hyperpower" is sometimes used for tetration, so I guess you could say "$n$ to the second hyperpower".

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There is no common term for $n^n$. I think a reasonable term if needed for casual use would be super square based on:

  • superexponentation is an alternative name for tetration
  • super-root is a name for the inverse of tetration (along with super-logarithm)
  • super square-root is the name for $x$ in the equality $y=x^x$
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