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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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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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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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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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1answer
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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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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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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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62 views

Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
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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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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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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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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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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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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
48 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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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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1answer
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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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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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Factors of a sequence resulting from repeated exponentiation

I have a sequence $a_n$: $1,2, 2^2, 2^{2^2}, 2^{2^{2^2}}, ...$ I would like to know how to factor $b_i = a_i-a_{i-1}$ where $a_0=1$ All I've been able to figure out so far is that 1 + $\sum\...
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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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1answer
101 views

How is tetration read in spoken English?

How would one read a tetration operation like $^4 3$ in spoken English? Meaning, what's the equivalent to reading $3 \times 4$ as "three times four" or $3^4$ as "three to the power of four" for ...
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1answer
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How to solve $x^x=a$ and related equations? [duplicate]

How can I solve the equation for $x$ when $x^x=2$ or any other constant? And is solving $x^{x^x}=a$ or $x^{x^{x^x}}=a$ or equations such as these even possible? What are these equations even called? ...
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Regularization of Exponentially exponential series?

Question What are the convergence properties of the last equation: $$K = e^{x} + x + \ln{x} + \ln\ln(x) + \dots $$ Can one artificially choose a value of $\ln (x)$ (since there is more than one ...
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1answer
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How would I go about solving for $x$?

How would I approach this problem? (Solving for $x$) $$x^{x}=e^{\Omega}$$ I tried using logarithms and rearranging, but it didn't seem to help: $$x=e^{\frac{\Omega}{x}}$$ $$\ln(x)=\frac{\Omega}{x}$$ $$...
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Tetration of $\pi$: Can it be a prime number?

Given the tetration as \begin{align} {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n \end{align} and the set of prime numbers as $\mathbb{P}$. Can you prove or to disprove the following ...
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1answer
142 views

Is there an exact value to the minimum of the infinite tetration of $x$?

Is there an exact value to the minimum of the function $$f_k(x)=\underbrace{x^{x^{x^{.^{.^.}}}}}_{2k\,\text{times}}$$ as $k\to\infty$, where $k=1,2,3,\cdots$? This visualisation in Desmos shows that ...
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1answer
131 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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What does $ ^3 3$ mean?

I was taking a test and this was a question. It does not mean $3^3$, nor is it a typo. The superscript was before the $3$ and I have no idea what it means. I tried researching but couldn't find an ...
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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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tetration as an iterative solution for the transcendental equation $\sqrt[x]{x}=y$

for positive integer $n$ use the notation $y^{[n]}$ to represent the $n$-th tetration of $y$, so $y^{[1]}=y$, $\, y^{[2]}= y^y$, $\,y^{[3]}=y^{y^y}$, and so on. a few simulations suggest that on $(0,...
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$f(exp(z)) = f(z) \cdot g(z) $ with closed form solutions?

Are there closed form solutions for functions $f(z),g(z)$ Such that $A)$ $$ \frac{f(exp(z))}{f(z)} = g(z) $$ $B)$ $g(z),f(z)$ are both analytic and nonconstant. $C)$ $g(z)$ is analytic at $z$ ...
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1answer
134 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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2answers
88 views

What is the value of $i^{i^{i^{\cdots}}}$ [duplicate]

If i can find the the value of the expression in lhs. Then i can find the correct option. But i am unable to find the value of expression on lhs which is i^i^i^i^... Upto infinity . How to find that. ...
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1answer
52 views

iterative solution of $x^x=a$

given $a,x \in (1,\infty]$then $x$ and $\sqrt[x]{a}$ are different numbers, except for a single value of $x$ which satisfies: $$ x^x = a $$ to solve this equation, therefore, it might help to look at ...
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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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0answers
85 views

$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ asymptotic?

Consider for positive real $x$ : $$ f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}} $$ How does this function behave ? How fast does it grow ? Faster than any fixed iteration of exp sure, but ...
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$ \DeclareMathOperator{\arcsinh}{arcsinh}w =\arcsinh( 1 + 2 \arcsinh( 1 + 2^2 \arcsinh ( 1 + 2^{2^2} \arcsinh( 1 + 2^{2^{2^2}} \arcsinh( 1 + \dotsm$

In the context of positive reals consider $$ w= \arcsinh( 1 + 2 \arcsinh( 1 + 2^2 \arcsinh ( 1 + 2^{2^2} \arcsinh( 1 + 2^{2^{2^2}} \arcsinh( 1 + \dotsm $$ Now consider a real $A > w$ Then the ...
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2answers
278 views

Is there a Tetration Function?

With exponentiation, you can raise numbers to complex, irrational, etc. This is defined as such: $$\exp(x)=\sum_{n=1}^\infty{x^n\over{n!}}$$ With $e=\exp(1)$ Is there some equation that would allow ...
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1answer
120 views

Why is $\underbrace {i^{i^{i^{.^{.^{.^{i}}}}}}}_{n \ times}$ converging? [duplicate]

Introduction: Recently I found out that $i^i \approx 0.20788$ has no imaginary part. I got interested and then wanted to know whether there are other $n$ for which $\underbrace {i^{i^{i^{.^{.^{.^{i}}}...
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0answers
68 views

How to solve $ax^x+bx+c=0$?

How can I solve $$ax^x+bx+c=0$$ or $$ax^{x^x}+bx^x+cx+d=0$$ where $x^x$ and $x^{x^x}$ - tetration? Is there analogue of discriminant for it?
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1answer
152 views

Indefinite Integral of $ \ln(x)^{\ln(x)^{\ln(x)^{.^{.^{.^{\ln(x)}}}}}}$ for an $ n $-number of $ \ln(x) $'s with respect to $ x $

This is tetration question about finding the indefinite integral. I am not sure where to start so any help would be appreciated. $$ I= \int \ln(x)^{\ln(x)^{\ln(x)^{\cdot^{\cdot^{\cdot^{\ln(x)}}}}}} ...
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General formula of first order derivative of the nth tetration of ln(x). [closed]

A follow-up question to the previous question. Is there a general formula of the first order derivative of $$ \ln(x)↑↑n$$ ? Where the $n$ is a constant independent of $x$.
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2answers
129 views

Magnitude of $f_3(n)$ compared to power towers of tens

In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$ The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$ For example, to calculate $f_3(5)$, we have to ...
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2answers
167 views

Quickly show that $\Bigg|(-1/2)^{(-1/2)^{(-1/2)}}\Bigg|=e^{\pi\sqrt{2}}$

Question: Is it true and can we quickly show that $$\Bigg|(-1/2)^{(-1/2)^{(-1/2)}}\Bigg|=e^{\pi\sqrt{2}}$$ Her was my solution. Abusively I write $a/b$ for the fraction ${a \above 1.5pt b} $. I write ...
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197 views

Shortcut to $x\uparrow \uparrow n$ for very large $n$ and $x\approx e^{(e^{-1})}$?

If the number $x$ is very close to $e^{(e^{-1})}$ , but a bit larger, for example $x=e^{(e^{-1})}+10^{-20}$, then tetrating $x$ many times can still be small. With $x=e^{(e^{-1})}+10^{-20}$ , even $x\...
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1answer
48 views

How slow does the iteration $x_1=r$ , $x_{n+1}=r^{x_n}$ converge for $r=e^{-e}$?

The number $r:=e^{-e}$ is the smallest number for which the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges. In other words, the iterarion $x_1=r$ , $x_{n+1}=r^{x_n}$ converges. ...
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1answer
160 views

Does $\lim_{n\rightarrow \infty} \left(r+\frac{1}{n^2}\right)\uparrow \uparrow n=e$ hold?

Does $$\lim_{n\rightarrow \infty}\left (r+\frac{1}{n^2}\right)\uparrow \uparrow n=e$$ hold ? $r$ is the number $e^{e^{-1}}$ , the largest real number for which the infinite power tower $r\uparrow r\...