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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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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

Problem picture Picture of given ackermann function So far, I have $$A(3,n) = A(2,A(3,n-1))$$ and then using $$A(2,n) = 2 \uparrow \uparrow n$$ I arrive at. $$A(3,n) = 2 \uparrow \uparrow A(3,n-1)$$ ...
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23 views

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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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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140 views

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
164 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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40 views

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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32 views

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
89 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
82 views

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
77 views

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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1answer
102 views

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
135 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
120 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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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
133 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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402 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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0answers
83 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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206 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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62 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
149 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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3answers
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Differentiation of $ \ln(x)^{(\ln(x)^{\ln(x)})}$ [closed]

Looking at a research paper that is looking at tetrations of ln(x) and their first order derivatives. I just want someone to provide a method of how to find the derivative of $$ \ln(x)^{(\ln(x)^{\ln(...
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2answers
127 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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159 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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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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0answers
70 views

Divisibility of difference of tetrations

A friend proposed to me this problem : Prove that for all naturals $n\ge 2$ we have : $n$ divides the number $2\uparrow \uparrow n - 2\uparrow \uparrow(n-1)$ here $2\uparrow \uparrow n := 2^{2^{{....
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1answer
46 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
158 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\...
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2answers
105 views

Has the equation $r^x\cdot \ln(r)=\ln(x)$ for $r<e^{-e}\ \ $ $3$ solutions?

This question is closely related to Is it hard to find out for which $r$ the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges? Let $r$ be a real number satisfying $0<r<e^{...
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1answer
175 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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0answers
60 views

Is it hard to find out for which $r$ the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges?

It is well known that the infinite power tower $$r\uparrow r\uparrow r\uparrow\cdots $$ with $r>0$ converges if and only if $e^{-e}\le r\le e^{1/e}$. I tried to prove it and I got stuck in the ...
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0answers
29 views

Number of steps to get an error less than $\epsilon$?

The iteration $$x_1=r$$ $$x_{n+1}=r^{x_n}$$ with $r=e^{e^{-1}}$ tends to $e$. What is the smallest index $n$ such that $|x_n-e|<\epsilon$ ? For small $\epsilon$, it seems that the smallest ...
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2answers
194 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 ...
2
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1answer
47 views

How can one simplify $\frac{\ln(4)}{W(\ln(4))}$ to get $2$?

The inverse of the equation $y=x^x$ is $\frac{\ln(x)}{W(\ln(x))}$. It is clear that the answer to $x^x=4$ is $2$, but the expression $\frac{\ln(4)}{W(\ln(4))}$ is not evidently equivalent with $2$. ...
6
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1answer
132 views

Is tetration and quintation etc. of infinite cardinals well defined?

I would like to know whether tetration and quintation functions are well defined for infinite cardinals, thus, for example, $$\aleph_0 \text{ [tet] } \aleph_0 = \aleph_0 ^ {\aleph_0 ^ {\aleph_0^{\...
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0answers
40 views

Tetration with 0 < #s < 1?

When I try numbers between 0 and 1 on my calculator app n-calc on my phone .. I get rapid convergence when the numbers are close to 1 .. And alternating but slow convergence when numbers are close to ...
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0answers
113 views

function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$

Here it's cited: the existence of the holomorphic function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$ had been demonstrated in 1950 by Hellmuth Kneser. ...
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0answers
55 views

$i_a = (i_b + i_c + 1)/2$ and $f(x) = f(x - f(x-1)) /2$?

Consider the set $i_0,i_1,...$ defined as $i_0$ is the smallest element and $i_0 = 0$. If $ (i_a - i_b)^2 < 1$ then $i_c$ is Also in the set and given by $$ i_c = (i_a + i_b + 1)/2$$ [*] Let $T(...
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74 views

Series of the form $\sum_{n=1}^\infty \frac{1}{ ^{s}n}$ where $^{s}n$ is the tetration of $n$

I am interested in the function $$T(s)=\sum_{n=1}^\infty \frac{1}{ ^{s}n}$$ where $s$ is an integer and $ ^s n$ is $n^{n^{n^{...}}}$ $s$ times. I know these series are convergent since tetration of $n&...