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Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

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Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
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Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than $...
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Fractional Composite of Functions

I would like to know how I can calculate a fractional composition of a function. Let be $f(x)$, where $x \in R$ and $f(x) \in R$. I now how to do $f(f(x))=f^2(x)$. Now suppose I would like to do $f^{\...
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Can tetration 'escape' the complex plane?

Considering subtraction can break out of the natural numbers and into integers, division can break out of integers and into rational numbers, and exponentiation can break out of rational numbers and ...
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Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
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What is the name for this: “x^^y” (as in 2^^2=2^2 and 3^^3=3^3^3 and so on)

How do you call / is there any specific name for the following: x^^y e.g.: 2^^2 = 2^2 3^^3 = 3^3^3 4^^4 = 4^4^4^4 ...
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Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
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Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
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Smooth Elementary Function that Outgrows All Tower Functions?

This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer. Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth. Is ...
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Tetration induction proof

The triple arrow-up denotes power towers in which the number of levels themselves is a power tower with a number of levels that is a power tower, and so on. For example, $$\begin{align} a\uparrow\...
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$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
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Comparison of the Sizes Arbitrarily Generated Compound Exponential Numbers

I've been wondering about how it might be possible, given two compound exponential numbers generated by recurrence schemes equipped with addition multiplication and exponentiation (to some integer ...
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Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
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half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and $x&...
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Can we obtain an explicit and efficient analytic interpolation of tetration by this method?

I am curious about this. It has been a very long time since I have ever toyed with this topic but it was an old interest of mine quite some time ago - maybe 8 years (250 megaseconds) ago at least, and ...
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1answer
398 views

How to solve 2 tetrated 0.5 times?

I've been really interested in tetration lately. So I came up with a seemingly simple problem to solve, which is 2 tetrated 0.5 times, which I'll write as the following. 2^^0.5 To make sense of this ...
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An Integral Designed to be on the Very Cusp of Convergence

Say we have an integer $n$∊ℕ₀ & a sequence of $n+1$ real numbers $\alpha_k\in[0,\infty)∀k$, where $k=0\dots n$, and using $\log^{[k]}$ to denote $k$ functionings of the logarithm ($\log^{[0]}x\...
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4answers
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Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher order ...
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Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$ The two smallest are $6771419$ and $72153167$, which I found by trial division. The number ...
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Tetration of non-integers: is there something wrong with this approach?

I'm trying to figure out a formula for tetration that will work for non-integer heights. I know the usual recurrence relation for tetration ($x \in \mathbb{R}, \text{ }n \in \mathbb{N})$: $${^{n}x} =...
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Is There an Operation Between Exponentiation and Tetration?

The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as \begin{align*} \text{exp} (a, n) = a^n=a\cdot a\cdots a \end{align*} and ...
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The Infinite Tetration of x? [duplicate]

The infinite tetration of ${x}$ basically means that we take a value and we continue to raise the value to its power forever. If that sounds confusing, it can be thought of as infinite repeated ...
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quadratic tetraic equation

I have recently found interesting thing: Lambert W funtion inverse of $f(x)=xe^x$ it was easy to find roots of $x^x=a$ but I wonder is possible to find formula for root of equation $ax^x+bx+c=0$ ????
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Larger value with right associative tetration?

Given right associative tetration where: $^{m}n =$ n^(n^(n^…)) And a situation such as: $^{m}n = y$ $^{q}p = z$ What is a practical way to calculate which of $y$ and $z$ are larger? I'm ...
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Taylor series for tetration

I would like to know what is known about Taylor series for tetration (and other hyper-exponentiations). Surprisingly, such information is rare on internet. Numerical values for expansion in ...
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Problem with derivative of $x^{x^x}$

I was recently watching blackpenredpen’s video (found here: https://m.youtube.com/watch?v=UJ3Ahpcvmf8) where he found the derivative of the the function $y = x^{x^x}$. Before watching the video, I ...
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Function $f$ s.t. $\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$

The questions are: 1) Does there exists some function $f$ s.t. $\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$ and $\lim_{x\to\infty}f(x)=\infty$? 2) Is $\big(\sum_{k=n}^{2^n}a_k\big)_n\to0$ is ...
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Solve $x=d^{d}\log(d)$ using Lambert $W(x)$

We have $a=b^b$, so $$\log(a)=b\log(b)$$ $$x=\frac{x}{W(x)}\log\left(\frac{x}{W(x)}\right)$$ $$b=\frac{\log(a)}{W(\log(a))}$$ Next we have $c=d^{(d^{d})}$, so $$\log(c)=d^{d}\log(d)$$ In general $^{k}...
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Maximum of exponential tower

First I introduce a notation similar to $\sum_{i=1}^n a_i$ for exponentiation. I.e. for any (potentially infinite) sequence $a_i$ we define $$ ES_{i=1}^n a_i = \left\{\begin{matrix} a_1 &...
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1answer
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Finding a function satisfies $\ln F(x+1)=a F(x)$

I want to find a smooth function $F$ satisfies $\ln F(x+1)=a F(x),\ x\in[0,2]$ and $F(0)=1$ I didn't prove the existance of the function but I think it exists. I can easily get $F(1)=e^a$ and $F(2)=e^...
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1answer
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Concise notation for iterated exponentiation involving an unknown

I am working with some tetration problems, such as below: $$y = e^{e^x}$$ and I am looking for a concise notation for this. In particular, I would like a way to indicate $n$ iterations of the ...
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Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^...}}}$ does not converge, that is $|W(-\...
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Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
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Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{...
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2answers
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How to compute the upper (repelling) fixpoint for $b^x=x$ ($1<b<1.44$) - using the LambertW?

For the computation of the lower (attracting) fixpoint $L_0$ for $b^{L_0} = L_0 $ (where the base $b$ is in the range for convergence $1 \lt b \lt e^{1/e}$) there is the simple formula $$L_0 = \exp(-W(...
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1answer
130 views

Area under the infinite tetration curve

What is the area under the curve where the infinite power tower converges? $$\lim_{y \to \infty} = {}^y x.$$ The formula for this curve is given by various sources as: $$\frac{\mathrm{W}(-\ln x)}{-\...
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1answer
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Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, ...
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How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\...
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0answers
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Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Knuth arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)\\f(x,n)=x\...
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1answer
172 views

Solve $i^{i^{i^\ldots}}$ [duplicate]

How to find $$i^{i^{i^\ldots}} \quad :\quad i=\sqrt{-1}$$ I'm able to find the solution for the finite powers using $$i=e^{i(2k\pi+\frac{\pi}{2})}\quad:\quad k\in\mathbb{Z}$$ $$i^{i}=e^{-(2k\pi+\...
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1answer
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What is the value of $i^{i^{i^\ldots}}$? [closed]

What is the value of $i^{i^{i^\ldots}}$? My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg ...
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0answers
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Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed?

Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper): $$ \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...
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2answers
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Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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1answer
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Is there a mathematical term, practical application, or area of math that covers a function raised to itself?

Some abstract examples would be: $f(x)^{f(x)}$ or $f(x)^{f(x)^{f(x)...}}$ Actual equations I've attempted to look at can be viewed here on desmos.com There seems to be a pattern of common convergence, ...
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0answers
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New/useful method for summation of divergent series?

Questions $$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$ Also obeys (see background for argument): $$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
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Sum of the reciprocal of tetration?

Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth. What are f(2) and f(3), and could you please also explain how you reached these answers?
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1answer
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Is it true that $\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$ has period $18$ and can never take the values $3$ and $6$?

Is it true that the arithmetical function $f:\mathbb{N}\setminus\{1\}\rightarrow \mathbb{Z}_9$ given by $$f(x\mid k)=\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$$ has period $...
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1answer
409 views

What is $\underbrace{2018^{2018^{2018^{\mathstrut^{.^{.^{.^{2018}}}}}}}}_{p\,\text{times}}\pmod p$ where $p$ is an odd prime?

This recent question inspired me to explore values concerning modulo arithmetic of tetrations, and I thus pose the following question. Is there a general expression for the value of $$\underbrace{...
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2answers
201 views

How to determine $n$, such that $x\uparrow \uparrow n>10^{100}$?

If $x$ is a real number greater than $e^{e^{-1}}$ , then $x\uparrow \uparrow n$ (A power tower of $n$ $x's$) tends to $\infty$, if $n$ tends to $\infty$. Therefore, there must be a number $n$, such ...