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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28 views

Modular Arithmetic and repeated exponentiation

I was messing around with mod and repeated exponentiation and noticed that if we let $P_n(k)$ denote repeated exponentiation by $n$, $k$ times then, $$\text{mod} \ b : a^{P_n(k)} \equiv a^{P_n(k-1)} \...
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Exploring extentions of Tetration

Recently I've been kind of curious about tetration, specifically why it doesn't introduce any new inverse functions in the way lower operations do- addition needs subtraction, multiplication needs ...
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How can we show that $\int_0^1 x^{x^{x^{{\cdot^{\cdot^{\cdot}}}}}}\ \mathrm{d}x = \frac{\pi^2}{12}$ [duplicate]

How can we show that $$I \equiv \int_0^1 x^{x^{x^{{\cdot^{\cdot^{\cdot}}}}}}\ \mathrm{d}x = \frac{\pi^2}{12}$$ I have only seen one approach to this by using the fact that $$\mathrm{W}(x) = \sum_{n=1}...
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How to find examples of periodic points of the (complex) exponential-function $z \to \exp(z)$?

Background: By considering the question which asks whether a certain summation-method $\mathfrak M$ for the (extremely divergent!) sum $\mathfrak M: S(z)=z + e^z + e^{e^z}+e^{e^{e^z}} + ...$ might be ...
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Tetration value for x=0 [duplicate]

I came across a video about the function $f(x) =x^{x^{x^{x^{...} }} } $ and I played around with GeoGebra trying to graph it. I've observed that if the number of x's is odd, then $f(0)$ is always ...
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25 views

Infinite tetration for complex domain [duplicate]

$$f(z)=z^{z^{z^{.^{.}}}}$$ $$for\space z \space \in C\rightarrow C$$ Can we define the range of this function(convergence-divergence specifically) without taking help from fractals? I checked out ...
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If $r\in\mathbb{Q}\setminus\mathbb{Z}$ is it possible that $r^{r^{r^r}}\in \mathbb{Q}$?

It's straightforward to prove that $r^r\notin\mathbb{Q}$, which furthermore allows us to use the Gelfond-Schneider theorem to prove that $r^{r^r}\notin\mathbb{Q}$. Is it true that $r^{r^{r^r}}\notin\...
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Is there a way to simplify the solution to $\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{x^{…}}}} dx$

My result for this integral is as follows: $$\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{....}}} = (e^{\frac{1}{e}})e - e - \frac{1}{2} - \sum_{k=1}^{\infty} \left( \frac{\gamma((k+2),(k))}{{k}^{(k+2)}\Gamma(...
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What would be the solution to x^^x= i? (imaginary unit)

(I don’t know proper math format sorry) I was looking at reverse operators. Let’s start with addition (reverse is subtraction) 1-2 is -1: so we extend our number field. Division next. We divide 3 by 2 ...
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Tetration: Summation of $\displaystyle {1 \over{}^{n}2}$

I'm going to get straight to point with this question - Can you find a closed form solution to this sum. $$\sum_{n=1}^\infty \displaystyle {1 \over{}^{n}2}$$ (where ${}^{n}2$ represents the nth ...
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Find the last digits of $a_{2009}$, and of $b_{2009}$.

Define the sequences $a_1, a_2,...$ and $*b_1, b_2,...*$ by $a_1 = b_1 = 7$ and $$a_{n+1} = {a_n}^7, \\ b_{n+1} = 7^{b_n}$$ for $n\ge 1$. Find the last digits of $a_{2009}$, and of $b_{2009}$. What ...
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Integrating $x^{x^x}$

Although one cannot find an elementary antiderivative of $f(x)=x^x$, we can still give a series representation for $\int_0^1 x^x dx$, namely: $$I_1=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^n}=0.78343\...
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Estimates on growth of $^{n}3$

I was dealing with a problem on tetration and am supposed to explain why this problem was challenging to me- obviously, difficulties stemmed from the amazing growth of $^{n}3$. The question now is: Is ...
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Confusing pictures about tetration !? [closed]

On the webpage http://tetration.org/Tetration/index.html, We are supposed to get an explanation of tetration, whatever that means exactly. In particular I feel the pictures are not well explained. ...
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what does $\lim _{x\to 0}$ $x^{x^{x^{x^{x^{x\cdots}}}}}$ evaluate to? [duplicate]

I was wondering if there is any possible way to define a limit for this form? I tried using L’hopital rule and tried evaluating the limit: $$\lim _{x\to 0} x^{x^{x^{x^{x^{x\cdots}}}}}$$ $$\lim _{x\...
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How to compute $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!\!\bmod 46,$ for power tower height $2020$?

What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$? The level of powers is $2020$. First there is no parenthesis so it means 3 power of 3 which is also power 3 ...
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What will be the domain of this tetration tower? [duplicate]

Let us consider a function: $f(x)= x^{x^{x^\cdots}}$ what will be the domain of this function. Like $f(1)=1$, $f(\sqrt2)=2$, but $f(2)$ will reach out to infinity. So, what is the domain of $f(x)$ ...
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New commutative hyperoperator?

After reading about Ackermann functions , tetration and similar, I considered the commutative following hyperoperator ? $$ F(0,a,b) = a + b $$ $$ F(n,c,0) = F(n,0,c) = c $$ $$ F(n,a,b) = F(n-1,F(n,a-...
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Evaluate $\lim\limits_{n\rightarrow\infty} \mathrm{srt}_n\left({^{n+1}}2\right)$

Notation: ${^n}x = x^{x^{\cdots^x}}$ is tetration, i.e. $x$ to the power of itself $n$ times. $\mathrm{srt}_n(x)$ is the super $n$-th root, or the inverse function of ${^n}x$, which is well defined ...
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Show that $\int\limits_0^1 \left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{⋰}}}}\ \mathrm{d}x=\frac{\pi^2}{12}$.

How can it be shown that $$\lim_{p\to\infty}I(p)= \lim_{p \to \infty}\int^{1}_0 (x^x)^{\scriptscriptstyle {(x^x)^{(x^x)^{(x^x)^{(x^x)^{(x^x)...(p \; times)}}}}}} dx= \frac{\pi^2}{12}$$ $I(1)=\...
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Computation of Digits in Tetration [duplicate]

According to Wikipedia, $^44$ has $8.1 \cdot 10^{153}$ digits. How can I calculate the number of digits for an arbitrarily large tetration, such as $^{11}11$? Thank you!
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Behaviour and closed forms of iterated functions

I'm interested in the behaviour of applying the same function repeatedly or oscillating between applying two different functions repeatedly. Let me explain. If I wanted to know what happens when I ...
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Why is $x^{1.36^x}$ such a good approximate to $\int_{0}^{x}t^t dt$?

So, once again I was experimenting on Desmos and found that $\int_{0}^{x}t^t dt$ can be approximated pretty well by the function $x^{1.36^x}$. It roughly becomes more accurate as $x$ approaches to ...
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Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
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Verifying uniqueness of my tetration

Previous two posts: Numerical instability of an extended tetration Verifying tetration properties Update: The first link only verifies continuity on $\mathbb R$, and so continuity cannot be used for ...
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2 different limits for $x^x$ - Could they somehow be applied?

So, I was just exploring the limit definition of e and seeing what I could create from it and after some time I landed at these 2 approximations for the power tower function - "$x^x$". $ e(x^{x-1}+(...
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Verifying tetration properties

In my previous question I asked about the numerical instability and convergence of my tetration. It would seem to be the case that it converges, but suffers from catastrophic cancellation. The ...
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Non integer Hyper-powers. [duplicate]

If I have a function $y=x^x$, that can be denoted in hyperpower notation as $^2x$, but I will be denoting it as $ y= $ hyp$_2(x) $. In general, for hyperpowers, $y=x^{x^{x^{...}x}}$ or in my notation $...
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Numerical instability of an extended tetration

For bases $a\in(1,e^{1/e})$, ${}^na=a^{({}^{n-1}a)}=a^{a^{a^{.^{.^{.^a}}}}}$ converges to a value denoted as ${}^\infty a$. By observing the convergence rate of this sequence, we can derive the limit: ...
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How can I find the value of a non-repeating exponent tower?

There are ways to express the function $$f(x)=x^{x^{x^{x^{x^{x^\cdots}}}}}$$ with $f(x)=\dfrac{W(-\ln(x))}{-\ln(x)} $ for other function like this; $$g(x) = x^{-x^{x^{-x^{x^{-x^{x^\cdots}}}}}} $$ I ...
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Find $\lim_{x\rightarrow 0}x^{x^{x^x}}$ [duplicate]

I already known how to prove that $\lim_{x\rightarrow 0}x^{x^x}=0$ and $\lim_{x\rightarrow 0}x^x=1$. I also tried to use L'Hôpital's rule for this question but it didn't work. How to find the limit? (...
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Find the maximum of $x^{x^{x^{⋰}}}.$

Question: Find the maximum of $x^{x^{x^{⋰}}}.$ Let $y = x^{x^{x^{⋰}}}.$ Then \begin{align} y & = x^y \\ \Rightarrow \ln y & = y\ln x \\ \Rightarrow \frac{1}{y} \frac{dy}{dx} & = y\left(\...
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Tetrating by non-integers? [duplicate]

Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (x[4]x), which is the same as x pentated by 2 (x[5]2), would look like on a graph. To do this, ...
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Rightmost decimal digits of Graham's number

How to find rightmost $n$ decimal digits of Graham's number efficiently. The last 500 digits are on the wiki/Graham's_number, but I want to know more. PowerTowerMod seems to be able to do it but is ...
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How to find the maximum arc length of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ and the value of $r$ at which it occurs?

After seeing a discussion about graphs of the relationship $x^x + y^y = r^r$, it got me interested in attempting to see what the graphical appearance of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ would ...
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How to express the infinite power tower $a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$?

Just a relatively simple question; I'm just wondering what would be the proper notation to use to express an infinite power tower that has each repeated exponent increasing by a value of $1$, like ...
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Is $\infty$ the solution to ${x}^{{x}^{{x}^{{x}^{x\dots}}}} = i$?

I started off trying to solve the problem ${x}^{{x}^{{x}^{{x}^{x\dots}}}} = i$ where $i$ is the imaginary unit, and with an infinite amount of $x$'s. I then substituted the $x$'s in the exponents as $...
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Prove $c,c^c,c^{c^c},c^{c^{c^c}},\ldots \pmod p$ with $p$ prime has period $1$ or $2$

Suppose $p$ is a prime number and $c$ is some constant value which is coprime to $p$. I found that $c,c^c,c^{c^c},c^{c^{c^c}},\ldots \bmod p$ have period $1$ or $2$. In other words, it seems ...
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For which values of reals $ a$ and $ x$ this:${x}^{x^{x^{x^…}}}=a$ could be work? [duplicate]

What happens to the W-Function in the expression ${x}^{x^{x^{x^...}}}=a$ and what values for $a$ and $x$ they take ? it seems there is a maximum value for $a$ that can take and which? Really I can't ...
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Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$

I'm a computer student, learning math just for fun. Today I was graphing for fun that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to ...
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139 views

Convergence of the sequence $x_{n+1} = a^{x_n}$ [duplicate]

Let $a > 0$. Show that the sequence defined by $$ x_0 = 1, \qquad x_{n+1} = a^{x_n} $$ converges for $a \leq e^{1/e}$. Any help is appreciated, I don't even know where to start with this. Edit to ...
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Finding the derivative of $x$ tetrated to the $x$

Differentiating the functions $x^x$, $x^{x^x}$ (or ${^2{x}}$ and ${^3{x}}$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this ...
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Nested Tetration properties

Regarding tetration, I know properties like ${}^a({}^bn)= {}^{ab}n$ do not hold in general. When $a=b=2$, for instance, we have $$ {}^2({}^2n)={}^2\left(n^n \right)=\left(n^n \right)^{\left(n^n \right)...
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60 views

The factors of a tetration plus an integer

There I was, just messing around with tetration, when I stumbled across this - $(x^x +1)/(x+1)$ = integer (for odd integer values of x) Playing some more with this it seems (not entirely sure as ...
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112 views

Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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95 views

Prove that ${}^ni$ is complex for all $n \ge 3$

We can define ${}^nx$ as $\underbrace{\displaystyle {x^{x^{\cdot ^{\cdot ^{x}}}}}}_{n\text{ times}}$ (Tetration). I conjecture that ${}^ni$ is complex for all $n \ge 3, n \in \mathbb{N}$. I've ...
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41 views

inverse tetrations of complex numbers

for the complex function $f_{(2)}(z)=z^z$,where in the complex plane does the inverse $z_{(2)}(f)$ not exist, same for inverse of functions $f_{(3)}(z)=z^{z^z}$ being $z_{(3)}(f)$ and so on for ...
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109 views

Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges?

let $f_1(x)=f(x)$ $f_2(x)=f(f(x))$ $f_3(x)=f(f(f(x)))$ and so on... Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges? It is okay if the ...
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137 views

Uniquely extended fractional iterations of $\exp$

Let us define the following basic conditions for an iterated exponential function: $$\exp^1(x)=e^x\tag{$\forall x$}$$ $$\exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$}$$ I then pondered what ...
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131 views

Compare power towers

Prove or disprove: $3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours. Taking logs is useless, and there seems to be no other way to compare. Thanks!

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