Questions tagged [power-towers]

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

Power towers are obtained by iterated exponentiation; an archetypal example is:

$\large a^{b^{c^d}}$

Power towers have been studied a lot; there is particularly many information on:

  • Modular arithmetic with power towers;
  • Convergence of "infinite" power towers; for example: $$\large\sqrt2^{\sqrt2^{\sqrt2^{\cdots}}} = 2$$

If we have a (finite) power tower with the same number repeated, such as the one with $\sqrt 2$ above, we speak of tetration. Tetration is the fourth hyperoperation, so if applicable, also include the tag.

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Attempt on fractional tetration

To start off, I was looking at the following ingeniously made form of the Gamma function: $$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$ which lies on the back of $$1=\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}$$ for all…
Simply Beautiful Art
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A problem in understanding infinite towers (tetration)

To solve equations involving power towers (infinite tetration) we usually do something like this: $$x^{x^{x^{x^{\dots}}}} =k$$ $$x^{(x^{x^{x^{\dots}}})} =k$$ $$x^k=k$$ $$x=\sqrt[k]k$$ But what if I do something like this: $$x^{x^{(x^{x^{\dots}})}}…
Renato Faraone
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Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each exponent level but different. I suspect this case is…
Zander
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Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using types of Incomplete Gamma functions. The goal is to…
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What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?

In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but I don't know how to solve this by hand.
shai horowitz
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computing ${{27^{27}}^{27}}^{27}\pmod {10}$

I'm trying to compute the most right digit of ${{27^{27}}^{27}}^{27}$. I need to compute ${{27^{27}}^{27}}^{27}(\bmod 10)$. I now that ${{(27)^{27}}^{27}}^{27}(\bmod 10) \equiv{{(7)^{27}}^{27}}^{27} (\bmod 10)$, so now I need to to compute…
Jozef
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Can $\int\limits_0^\infty e^{ix^ x}dx$ be written without a limit?

We know about the Fresnel Integrals: $$C(x)=\int \cos x^2 \, dx,\quad S(x)=\int \sin x^2 \, dx$$ which can also be written as: $$\int e^{ix^2}dx=C(x)+i\,S(x)$$ To make a more interesting and tetration based integral with a rapidly oscillating part…
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Does $\mathrm{\int W(ln(x))dx}$ have a closed form?

This is follow up to this question which you will have to see for context: Is there a better solution for $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(a))}{n^{n+1}}dt}$$ which had the closed form of…
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Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?

A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real positive numbers denoted by $\Bbb R^+$ and not from 0…
Tyma Gaidash
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Limits of negative-power tower

Consider the following: $$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$ It doesn't converge for $m \to \infty$, but eventually alternates between two values where the…
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Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think there will be an $N$ for which $$ 3 \uparrow \uparrow…
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Find a value of $\;\lim\limits_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$

Find a value of $$\lim_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$$ For your information$,\quad\uparrow\uparrow$ is a tetration defined as $$a\uparrow\uparrow…
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Taylor series of a power tower

I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by $$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$ where $B_n$ are the Bell Numbers. However, I can't figure out a Taylor series for the function $$\exp(\exp(\exp(x))) =…
Franklin Pezzuti Dyer
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Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
Taha Akbari
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Calculating the residue of power towers

I want to calculate the residue of a power tower. How do I do that? For example, I want to know the answer to this: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
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