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If $x \cdot 2 = x + x$

and $x \cdot 3 = x + x + x$

and $x^2 = x \cdot x$

and $x^3 = x \cdot x \cdot x$

Is there an operator $\oplus$ such that:

$x \oplus 2 = x^x$

and $x \oplus 3 = {x^{x^x}}$?

Also, is there a name for such a set of operators ops where...

Ops(1) is addition

Ops(2) is multiplication

Ops(3) is exponentiation

Ops(4) is $\oplus$

...and so on

Also, is there a branch of math who actually deals with such questions? Have these questions already been answered like 2000 years ago?

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    $\begingroup$ You may be interested in the Ackermann function. Makes for fun problems in recursion theory and such. $\endgroup$ – Jyrki Lahtonen Jul 5 '16 at 9:48
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    $\begingroup$ FYI $(x^x)^x = x^{(x^2)} \ne x^{(x^x)}$ so the operation is not associative so the notation ${x^x}^x$ is ambiguous. We usually take it to mean $x^{(x^x)}$ because $(((x^x)^x)^x).... = x^{(x^m)}$ and can be expressed that way. $\endgroup$ – fleablood Jul 5 '16 at 17:34
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    $\begingroup$ Also of note: the "Ops" are known as Hyperoperations. $\endgroup$ – mbomb007 Jul 5 '16 at 18:56
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    $\begingroup$ i think fleablood got the convention backward. $a^{b^c}$ is understood as $a^{(b^c)}$ because $(a^b)^c = a^{bc}$. $\endgroup$ – Anton Sherwood Jul 6 '16 at 0:52
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This operation ${\rm Ops}(4)$ is called tetration, from the greek root tetra meaning four; it's also sometimes called a "power tower". There are also many further generalizations of this type of sequence; Knuth's up-arrow notation gives $a^{a^{a^a}}=a\uparrow\uparrow4$, so that $a\uparrow\uparrow n$ is the tetration operation. By adding more arrows you get pentation and so on, and the Conway chained arrow notation generalizes this still further.

FYI, for "to the power of-ation" the word you're looking for is exponentiation.

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    $\begingroup$ Also that is similar to graham number. en.wikipedia.org/wiki/Graham%27s_number $\endgroup$ – Takahiro Waki Jul 5 '16 at 9:19
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    $\begingroup$ Aha, and the answer to the second question is hyperoperation: en.m.wikipedia.org/wiki/Hyperoperation $\endgroup$ – uzilan Jul 5 '16 at 12:19
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    $\begingroup$ Ah, the whims of the internet. This does not deserve to be my highest-voted answer. $\endgroup$ – Mario Carneiro Jul 6 '16 at 4:03
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    $\begingroup$ It's the bikeshed, Mario; you get used to it. $\endgroup$ – J. M. is a poor mathematician Jul 6 '16 at 12:36
  • $\begingroup$ The problem with this answer is that a↑↑n is not defined to be tetration operation. It is a consequence of a↑n being defined as $a^n$ and then further arrows is defined to be n number of iterations of itself minus one arrow on a. Much prefer @Curd 's answer $\endgroup$ – Ariana Jul 8 '16 at 17:59
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A more general function that combines all those operators has been defined by Ackermann:

$ \varphi(m,n,p) = \begin{cases} \varphi(m, n, 0) = m + n \\ \varphi(m, 0, 1) = 0 \\ \varphi(m, 0, 2) = 1 \\ \varphi(m, 0, p) = m &\text{ for } p > 2 \\ \varphi(m, n, p) = \varphi(m, \varphi(m, n-1, p), p - 1) &\text{ for } n > 0 \text{ and } p > 0. \end{cases} $

So for $p = 0, 1, 2$ you get

$\phi(m, n, 0) = m + n $
$\phi(m, n, 1) = m \cdot n $
$\phi(m, n, 2) = m ^ n $

and

$\phi(m, n, 3) = \overbrace{{{m ^ m} ^ m} ^ {...}}^{n}$

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It is known as a tetration, and it is normally written as $^na$ where n is the height of the power tower. It is the forth hyperoperation.

The zeroth hyperoperation is the successor function, and the first is the zeroth hyperoperation iterated, and so on

A more general way to define the nth hyperoperation is, using the notation, $H_n(a,b)$ where n is the nth hyperoperation,

${\displaystyle H_{n}(a,b)={\begin{cases}b+1&{\text{if }}n=0\\a&{\text{if }}n=1{\text{ and }}b=0\\0&{\text{if }}n=2{\text{ and }}b=0\\1&{\text{if }}n\geq 3{\text{ and }}b=0\\H_{n-1}(a,H_{n}(a,b-1))&{\text{if }n\in\mathbb{N},n>3}\end{cases}}}$

Some notations for hyperoperations are(for $H_n(a,b)$:

  1. Square bracket notation: $a[n]b$
  2. Box notation: $a{\,{\begin{array}{|c|}\hline {\!n\!}\\\hline \end{array}}\,}b$
  3. Nambiar's notation : $a\otimes ^{n-1}b$
  4. Knuth's up arrow notation: $a\uparrow^{n-2}b$
  5. Goodstien's notation: $G(a,b,n)$
  6. Conway's chained arrow notation: $a\rightarrow b\rightarrow (n-2)$
  7. Bowers exploding array function: $\{a,b,n,1\}$
  8. Original Ackermann function: ${\begin{matrix}\phi (a,b,n-1)\ {\text{ for }}1\leq n\leq 3\\\phi (a,b-1,n-1)\ {\text{ for }}n\geq 4\end{matrix}}$
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    $\begingroup$ Wow, Arianna Grande as a mathematician and stackexchange user. Exciting, $\endgroup$ – Chisko Jul 6 '16 at 17:26
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    $\begingroup$ @Cheskos Ariana $\endgroup$ – Ariana Jul 6 '16 at 17:40
  • $\begingroup$ Is any of these 8 extending Peter Hurfords $C(a,\dots)$ (see Extending the Extensions)? $\endgroup$ – Rudi_Birnbaum Mar 11 '18 at 21:47
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    $\begingroup$ @R_Berger The extension, I believe, grows faster than every function in the 8 listed $\endgroup$ – Ariana Mar 12 '18 at 10:24

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