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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?

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You might want to ask this in the Tetration Forum –  Pedro Tamaroff Feb 19 '12 at 5:13
    
FWIW, repeated exponentiation is called tetration. –  JavaMan Feb 19 '12 at 5:20
    
I know it as the tower function/operator. –  Raphael Feb 19 '12 at 12:05
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3 Answers 3

up vote 6 down vote accepted

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 $2\uparrow\uparrow n$.

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thanks a lot, that's what I needed –  Sergey Feb 19 '12 at 5:06
    
No problem, glad to help. –  Zev Chonoles Feb 19 '12 at 5:08
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Yes, using Knuth's up-arrow notation. In your case, $2\uparrow\uparrow n$.

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According to this definition you can define this number as :

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

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