2
$\begingroup$

I have this formula $$\underbrace{2^{2^{2^{.^{.^{.^{2^2}}}}}}}_n$$i.e. where the total number of 2's is $n$.

Is there any way to write it as a single mathematical function?

  • 0
    You might want to ask this in the [Tetration Forum](http://math.eretrandre.org/tetrationforum/index.php)2012-02-19
  • 0
    FWIW, repeated exponentiation is called [tetration](http://en.wikipedia.org/wiki/Tetration).2012-02-19
  • 0
    I know it as the *tower* function/operator.2012-02-19

3 Answers 3

6

Knuth invented a notation for these kinds of expressions, called "up-arrow notation".

To express the power tower in your question with up-arrow notation, we can simply write $2\uparrow\uparrow n$.

  • 0
    thanks a lot, that's what I needed2012-02-19
  • 0
    No problem, glad to help.2012-02-19
5

Yes, using Knuth's up-arrow notation. In your case, $2\uparrow\uparrow n$.

2

According to this definition you can define this number as :

$$^n2 = \begin{cases} 1, & \text{if }n=0 \\ 2^{[^{n-1}2]}, & \text{if }n>0 \end{cases}$$