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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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How to ompute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $ \int{x↑↑n} dx $ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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Has something like “Knuth's Up-Arrow Factorial Notation” ever been used? If so, what practical uses does it have?

I was studying Knuth's up-arrow notation and I was wondering if ever something like "Knuth's up-arrow factorial notation" has ever been used. Now I know this probably isn't a recognizable term, ...
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A new interesting pattern to $i\uparrow\uparrow n$ that looks cool (and $z\uparrow\uparrow x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to ...
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1answer
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Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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How to solve $x^x = y^y \mod p$?

Let $p>5$ be a prime. How to solve $x^x = y^y \mod p$? How many solutions are there for a given $p$ such that $x,y < p$? I know the discrete logarithm and the theory of quadratic residues, but ...
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How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
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Continuous, infinitely differentiable tetration map

We know that the exponential map is infinitely differentiable; is there a tetration map $TET$ defined on $(0,\infty)$ such that 1) $TET(x) > TET(y)$ when $x > y$ 2) $TET(e^x) = TET(x) +1$ 3) $...
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Unpoven generalized tetration integral on Wikipedia? [duplicate]

While I was investigating the interesting properties of tetration, I found the integral above. No matter where I search, I just couldn't find the proof for this. Could someone point me to a resource ...
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1answer
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Power tower(infinite tetration) [duplicate]

How to find the domain and range of $Y=x^{x^{x^{x^{x^{...}}}}}=x^Y$ I know how to differentiate the function. I don't know how to proceed further. We should prove that domain:$[1/(e^e),e^{(1/e)}]$ ...
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Pattern in power towers of 2 involving last digits

We have \begin{align} 2^{2^{2}} &\mod 10 = 6 \\ 2^{2^{2^2}} &\mod 100 = 36 \\ 2^{2^{2^{2^2}}} &\mod 1000 = 736 \\ 2^{2^{2^{2^{2^{2}}}}} &\mod 10000 = 8736 \\ 2^{2^{2^{2^{2^{2^2}}}}} &...
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Associocommutativity

One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring ...
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644 views

Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
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1answer
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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=\...
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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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5answers
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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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2answers
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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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1answer
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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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2answers
639 views

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
582 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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296 views

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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2answers
109 views

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
135 views

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
53 views

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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4answers
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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
150 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)}{-\...