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$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$.

$(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$.

Is there an operator, such that if $x^{x^{\cdot^{\cdot^{x}}}}$, i.e. $x$ raised to the $x$th power $n$ times (with right associativity), we can write something like $x$ ¤ $n$, where the generic currency sign ¤ is a placeholder for the correct operator, if it exists?

Does such an operator exist? If so, what is the correct symbol? If not, why?

Is there a name for this operation? If so, what is it?

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    You might try and look at wikipedia:hyperoperations2012-11-21
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    @Serkan I put the wrong variable; I meant to put $x$.2012-11-21
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    Of course it exists, because you just stated what it is. :)2012-11-21
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    First of all, what exactly does $x^{x^x}$ mean? The power operator is not associative: $$x^{\hat{}}(x^{\hat{}}x) \ne (x^{\hat{}}x)^{\hat{}}x \, .$$ For example, $3^{\hat{}}(3^{\hat{}}3) = 3^{27} = 7,625,597,484,987$ while $(3^{\hat{}}3)^{\hat{}}3 = 27^3 = 19,683.$2012-11-21
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    @FlybyNight $x$ ^ $(x$ ^ $x)$2012-11-21
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    Sometimes this $^yx$, notation is used for integer tetration, where $^3x=x^{x^x}$.2012-11-24

1 Answers 1

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Depending on who you ask, you're going to get different responses.

You are referring to what is known as tetration. You can thank Reuben Louis Goodstein for this word, which is roughly "four iteration". It has many notations:

$$ ^{n}a, a {\uparrow\uparrow} n,a \rightarrow n \rightarrow 2, \text{uxp}_{a}n, a^{\underline{n}}, a^{(4)}n, \text{hyper}_4(a,n), . . . $$

For more, see Wikipedia: Tetration Notation.