Potency tower

x ↑↑ a for a = 1,2,3,4, ...

In mathematics , especially number theory , one speaks of a power tower when the exponent of a power itself is represented as a power. This can then be repeated in that the exponent of the exponent is also a power and so on, so that the bases build up to a tower that is closed by the (last) exponent. The notation is commonly used for numbers where the exponent would be too large in normal notation, e.g. B .:

${\displaystyle 2^{1.267.650.600.228.229.401.496.703.205.376}=2^{2^{100}}}$

The larger the number, the clearer the advantage of this abbreviated notation becomes.

${\displaystyle 2^{2^{2^{2^{2^{2}}}}}=2^{2^{2^{2^{4}}}}=2^{2^{2^{16}}}=2^{2^{65.536}}}$

Even the exponent of this power of 2 would have, in decimal notation, 19,728 places. The overall result would then hardly be usable or understandable.

The convention applies that power towers are processed “from top to bottom”, ie starting with the highest power: therefore means and not .
${\displaystyle 2^{3^{4}}}$ ${\displaystyle 2^{\left(3^{4}\right)}=2^{81}}$ ${\displaystyle \left(2^{3}\right)^{4}=8^{4}=2^{12}}$

With the help of this notation, very large numbers can be clearly represented, which are quickly beyond any direct imaginability and which can no longer be represented in absolute length and as a simple power or only cumbersome.

However, there are numbers that are so large that even this notation is no longer sufficient to represent them. So if a power tower has too many levels to be represented, alternative notations such as the hyper operator are used .

Representation with sequences and infinite power towers

A finite power tower of the form (see also arrow notation )

${\displaystyle {\begin{matrix}x\uparrow \uparrow n&=&\underbrace {x^{x^{{}^{.\,^{.\,^{.\,^{x}}}}}}} \\&&n{\mbox{ Kopien von }}x\end{matrix}}}$

with and agrees with the -th link of the ${\displaystyle x\in \mathbb {R} ^{\geq 0}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle a_{n}(x)}$

${\displaystyle a_{n}(x):={\begin{cases}x,&{\text{falls }}n=1\\x^{a_{n-1}(x)},&{\text{falls }}n>1\end{cases}}}$

recursively defined sequence . This is called the partial tower sequence and identified with the infinite power tower (analogous to the term infinite series). ${\displaystyle (a_{n}(x))}$

If convergent with the limit value then the (infinite) power tower is called convergent with ${\displaystyle (a_{n}(x))}$ ${\displaystyle A(x),}$

${\displaystyle x\uparrow \uparrow \infty =x^{x^{{}^{.\,^{.\,^{.\,}}}}}=A(x).}$

Even Leonhard Euler has recognized that the tetration

${\displaystyle x\uparrow \uparrow \infty =x^{x^{{}^{.\,^{.\,^{.\,}}}}}\qquad {\text{f}}{\ddot {\text{u}}}{\text{r}}\;x\in \mathbb {R} ^{\geq 0}}$

converges if and only if

${\displaystyle 0{,}065988\approx {\frac {1}{e^{e}}}=e^{-e}\leq x\leq e^{\frac {1}{e}}\approx 1{,}444668.}$

The function thus defined

${\displaystyle A\colon [e^{-e},e^{\frac {1}{e}}]\to [{\tfrac {1}{e}},e],\quad A(x)=x^{x^{{}^{.\,^{.\,^{.\,}}}}}}$

is strictly monotonously growing and bijective . Its inverse function is given by

${\displaystyle A^{-1}\colon [{\tfrac {1}{e}},e]\to [e^{-e},e^{\frac {1}{e}}],\quad A^{-1}(x)=x^{\frac {1}{x}}}$.

See also

• Lambertian W function
• Skewes number
• Graham number
• Googol

Web links

• Eric W. Weisstein : Power Tower . In: MathWorld (English).

Individual evidence

1. ^ R. Arthur Knoebel: Exponentials Reiterated . In: The American Mathematical Monthly . tape 88 , no. 4 , April 1981, pp. 235-252 , doi : 10.2307 / 2320546 .