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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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1answer
37 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
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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
143 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
82 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
283 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
184 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
175 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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1answer
161 views

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
137 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
90 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
54 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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$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ 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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$w =\operatorname{arcsinh}(1+2\operatorname{arcsinh}(1+2^2\operatorname{arcsinh}(1+2^{2^2}\operatorname{arcsinh}(1+\dotsm$

In the context of positive reals consider$\DeclareMathOperator{\arcsinh}{arcsinh}$ $$ w= \arcsinh( 1 + 2 \arcsinh( 1 + 2^2 \arcsinh ( 1 + 2^{2^2} \arcsinh( 1 + 2^{2^{2^2}} \arcsinh( 1 + \dotsm $$ ...
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451 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
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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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76 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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153 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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1answer
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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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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
194 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 $$\Bigg|(-1/2)^{(-1/2)^{(-1/2)}}\Bigg|=e^{\pi\sqrt{2}}$$ Here was my solution. Abusively I write $a/b$ for the fraction ${a \above 1.5pt b} $. I ...
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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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1answer
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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
163 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
120 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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2answers
413 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
64 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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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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245 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 ...
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1answer
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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 $2$. ...
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1answer
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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
62 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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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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62 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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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&...
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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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3answers
398 views

Solutions of $a^{a^x}=x$ for fixed $a>0$

I am interested in the equation $a^{a^x}=x$ for some fixed $a>0$. Is there some way to rearrange for $x$ or solve otherwise? What about the nature of the solutions? For which fixed $a>0$ are ...
3
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1answer
704 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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0answers
227 views

Addition, Multiplication, Exponentiation, Tetration. Iterated functions.

Please note that I am exceptionally talented in over-complicating even the simplest of topics, however this may be worth a read. Also, the following operations are only performed with positive ...
1
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2answers
90 views

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(...
2
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0answers
230 views

$f ^{\prime \prime}(x) = f(x) f^{\prime \prime} (x-1) $

I know the equation $$ f^{\prime} (x) = f(x) f^{\prime} (x-1) $$ is solved by $f(x) = C$ or by tetration ( $ f(x+1) = \exp(f(x)) $). So I wonder What are the solutions to $$g^{\prime \prime}(x) =...
1
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1answer
42 views

Variation of Fermat number, looking for literature on special properties

Suppose we define a recursive function as follows: $F_k(0) = 1$ For $i \epsilon \mathbb{N}, i>0$, we define: $F_k(i) = k^{F_k(i-1)}$ So if $k=2$ we get: $F_2(3) = 2^{2^2}$ $F_2(4) = 2^{{2^2}^...
5
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2answers
742 views

Pentation Notation - How does it work? [duplicate]

When going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^3}}=...
2
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1answer
132 views

Analytically continuing the series exponentially exponential series?

Background I was recently toying with a series: $$ S = \exp(x)+ \exp(\exp(x))+ \exp(\exp(\exp(x)))+ \dots $$ Taking exponential both sides: $$ \implies e^S = \exp(\exp(x)) \cdot \exp(\exp(\exp(x)))...
2
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0answers
32 views

“Derivative-like” Operator such that $H(f(x)^{g(x)}) = H(f(x))^{H(g(x))}$

Given 2 functions $f(x),g(x)$ we have: $$D(f(x)+g(x)) = D(f(x))+D(g(x))$$ And $$D^*(f(x)g(x)) = D^*(f(x))D^*(g(x))$$ where $D^*$ is the multiplicative derivative: $$D^*(f(x)) = lim_{h\rightarrow 0} {\...