Mathematical function for the powers

Question

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?

You might want to ask this in the Tetration Forum – Pedro Feb 19 '12 at 05:13 — Pedro, Feb 19 '12 at 05:13
FWIW, repeated exponentiation is called tetration. – JavaMan Feb 19 '12 at 05:20 — JavaMan, Feb 19 '12 at 05:20
I know it as the tower function/operator. – Raphael Feb 19 '12 at 12:05 — Raphael, Feb 19 '12 at 12:05

score 6 · Accepted Answer · answered Feb 19 '12 at 05:03

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 $2\uparrow\uparrow n$.

answered Feb 19 '12 at 05:03

Zev Chonoles

score 5 · Answer 2 · answered Feb 19 '12 at 05:02

5

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

answered Feb 19 '12 at 05:02

Apprentice Queue

score 2 · Answer 3 · answered Feb 19 '12 at 05:10

2

According to this definition you can define this number as :

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

answered Feb 19 '12 at 05:10

Pedja

3 Answers3