# Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, 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?

• You might try and look at wikipedia:hyperoperations – Gottfried Helms Nov 21 '12 at 19:20
• @Serkan I put the wrong variable; I meant to put $x$. – ctype.h Nov 21 '12 at 20:40
• Of course it exists, because you just stated what it is. :) – Harry Altman Nov 21 '12 at 20:48
• 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 $(3^{\hat{}}3)^{\hat{}}3 = 27^3 = 19,683.$ – Fly by Night Nov 21 '12 at 21:03
• @FlybyNight $x$ ^ $(x$ ^ $x)$ – ctype.h Nov 21 '12 at 22:41

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