28
$\begingroup$

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?

$\endgroup$
4
  • 10
    $\begingroup$ You may be interested in the Ackermann function. Makes for fun problems in recursion theory and such. $\endgroup$ Jul 5, 2016 at 9:48
  • 1
    $\begingroup$ 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, 2016 at 17:34
  • 2
    $\begingroup$ Also of note: the "Ops" are known as Hyperoperations. $\endgroup$
    – mbomb007
    Jul 5, 2016 at 18:56
  • 3
    $\begingroup$ i think fleablood got the convention backward. $a^{b^c}$ is understood as $a^{(b^c)}$ because $(a^b)^c = a^{bc}$. $\endgroup$ Jul 6, 2016 at 0:52

3 Answers 3

55
$\begingroup$

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.

$\endgroup$
8
  • 6
    $\begingroup$ Also that is similar to graham number. en.wikipedia.org/wiki/Graham%27s_number $\endgroup$ Jul 5, 2016 at 9:19
  • 4
    $\begingroup$ Aha, and the answer to the second question is hyperoperation: en.m.wikipedia.org/wiki/Hyperoperation $\endgroup$
    – uzilan
    Jul 5, 2016 at 12:19
  • 6
    $\begingroup$ Ah, the whims of the internet. This does not deserve to be my highest-voted answer. $\endgroup$ Jul 6, 2016 at 4:03
  • 1
    $\begingroup$ It's the bikeshed, Mario; you get used to it. $\endgroup$ Jul 6, 2016 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, 2016 at 17:59
9
$\begingroup$

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

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

$\endgroup$
1
9
$\begingroup$

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: ${\begin{matrix}\phi (a,b,n-1)\ {\text{ for }}1\leq n\leq 3\\\phi (a,b-1,n-1)\ {\text{ for }}n\geq 4\end{matrix}}$
$\endgroup$
4
  • 1
    $\begingroup$ Wow, Arianna Grande as a mathematician and stackexchange user. Exciting, $\endgroup$
    – Chisko
    Jul 6, 2016 at 17:26
  • 1
    $\begingroup$ @Cheskos Ariana $\endgroup$
    – Ariana
    Jul 6, 2016 at 17:40
  • $\begingroup$ Is any of these 8 extending Peter Hurfords $C(a,\dots)$ (see Extending the Extensions)? $\endgroup$ Mar 11, 2018 at 21:47
  • 1
    $\begingroup$ @R_Berger The extension, I believe, grows faster than every function in the 8 listed $\endgroup$
    – Ariana
    Mar 12, 2018 at 10:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.