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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34 views

Average of all $a_n$ where $a_0$=1, $a_{n+1}=\omega^{a_n}$, and $\omega=\frac{\pi i}{\ln(2)}$

Essentially, I've noticed that tetrations of $\omega=\frac{\pi i}{\ln(2)}$ seem to converge on a cycle of three fixed points. Specifically, if $a_0$=1, and $a_{n+1}=\omega^{a_n}$, then we find $$n\...
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63 views

Is the infinite tower power $e^{e^{e^{e…}}}$ defined?

Here's what happens when I try to evaluate the infinite tower power, $e^{e^{e^{e...}}}$: $$x=e^{e^{e^{e...}}}$$ $$x=e^x$$ $$\ln x =x$$ $$\ln x=e^x$$ No Real solution. But come on! All those piled up e'...
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Power tower of infinite height mod n

So this one's been bothering me for a while and I can't figure it out Define $^kb$ as $b^{b^{b^{...}}}$ as the power tower of $b$ of height $k$ What I want to do is understand the behavior of $\lim_{k ...
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Tetration rule for differentiation.

For the people who doesn't know what tetration is:- $^na=\begin{cases}1&\text{if}\,n=0\\a^{a^{.^{.^{.^{a}}}}}&\text{if}\,n\in\mathbb{N}\end{cases}$ There are $n$ a's in that power tower. And ...
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Find all numbers $x$ and $y$ such as that $ ^yx \ =\ ^xy$.

Find all integers $x$ and $y$ such as that $ ^yx \ =\ ^xy$, where $^yx$ is a tetration. Can it also be solved in real numbers? I already know this: https://en.wikipedia.org/wiki/Tetration. I tried ...
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finding a discrete log analogous for tetration in finite fields

Tetration is defined as \begin{align} T(a, n) &= \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n\space times} \\ T(a, 0) &= 1 \end{align} Now, I was wondering if there is an efficient way to solve ...
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Is it possible to analytically solve this question

I came across this question which I successfully solved computationally, but I was wondering if there is an analytical way of doing it. Find Real and Imaginary parts of: $$ i^{i^{i^{i^{.^{.^ {2019}}}}...
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260 views

Possible real extension of tetration, or ones with similar growth rate; what makes it difficult?

This question arose when I read a (very introductory) googology book. Tetration is essentially repeated exponentiation (right-associative), just like how multiplication is repeated addition, defined ...
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Golden Exponent? Tetration

I read somebody say “golden exponent $ x^x=x+1 $” now I didn’t understand what he meant but it really fascinated me thinking about some type of tetration version of the golden number. A number with a ...
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Why does ${x}^{x^{x^{x^{\,.^{\,.^{\,.}}}}}}$ bifurcate below $\sim0.065$?

When you calculate what ${x}^{x^{x^{x\cdots }}}$ converges to between $0$ and $1$, before approximately $0.065$ the graph bifurcates. Why does this happen and is there a reason for it happens at that ...
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Given a real number what is the minimum number of times it can be tetrated to get an intiger?

I really enjoyed this video on the possibility that $\pi^{\pi^{\pi^\pi}}$ is an integer, but i thought that it was a case of a more interesting general problem. Given a real $x$ and an integer $y$, ...
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Reference for infinite exponentiation

I'm trying to find a book or a paper on infinite exponentiation: more precisely, it should be proving its full interval of convergence on the positive real line, i.e. if $x\in\mathbb{R}^+$, then $x^{x^...
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Complex Tetration

Does anyone know how to calculate complex tetrations? There are formulas for tetrations but I always end up in a complex tetration: $^{n} a = a^{(^{n-1} a)}$ Now, if we use this formula for complex ...
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Would this closed form derivative rule for tetration be correct for $n$ as a nonnegative integer? If so, can it be more concise? [duplicate]

For the recent week I've been diving myself into tetration out of pure curiosity, and after watching videos about how to find the derivative of $^2x$ and $^3x$, I decided that I wanted to look into ...
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65 views

Iterated logarithm to the $n - 1$ of the auto tetration n^^n

I'm considering the sequence $1$ $\log\left(2^2\right)$ $\log\left(\log\left(3^{3^3}\right)\right)$ $\log\left(\log\left(\log\left(4^{4^{4^4}}\right)\right)\right)$ I tried solving this using Python ...
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64 views

Is $x\uparrow\uparrow x$ PR?

By $\uparrow$ I'm referring to Knuth's notation. In particular, $$x\uparrow\uparrow x=x^{x^{\cdots^{x}}},$$ where exponentiation is iterated $x$ many times. I suspect not as I think it may grow faster ...
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77 views

tetration limit tending to 0

Let $x \uparrow \uparrow n = x^{x^{x^{x}}}$ ex: $x \uparrow \uparrow 3 = x^{x^x}$ I'm trying to evaluate the limit $$ \lim_{n \rightarrow \infty} \frac{3 \uparrow \uparrow \frac{1}{n} - 1 }{2 \...
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What is the largest finite number you can make using no more than 6 characters?

(I am new to the Math Stack Exchange community so tell me if this question is not allowed, however, I checked on Meta first.) (Also, I don't really know which tags I should assign this question, so ...
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What is the result of ${^{i}i}$? [duplicate]

$i+i=2i$, $i\cdot i=-1$, $i^i=e^{-\pi/2}$, but what is the result of ${^{i}i}$? That is, what is the $i$-th tetration of $i$?
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How are the hyperoperations defined?

For the first few operations such as addition and multiplication they follow rules such as $$A(x,y+1)=A(x,y)+1$$ $$A(x,0)=x$$ For multiplication $$M(x,y+1)=M(x,y)+x$$ $$M(x,1)=x$$ So for a general ...
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Finding the last $17$ digits of $1707 \uparrow \uparrow 1783$.

The objective is to find the last $17$ digits of : $1707 \uparrow \uparrow 1783$. In other words: $$ 1707 \uparrow \uparrow 1783 \pmod{10^{17}} $$ where $\uparrow \uparrow$ represents tetration or ...
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Find last three digits of $8^{8^8}$

I am attempting to find $8^{8^8}$ (which, by the way, means $8^{(8^8)}$) without any means such as computers/spreadsheets. Here's my attempt so far, and I'm pretty sure my answer is correct, but I ...
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63 views

Solving $a_{n+1} = x^{a_n}$ for various $x$ [duplicate]

(a) Consider a sequence defined by $a_0=1$ and $a_{n+1}=(\sqrt2)^{a_n}$ . Prove that limit exists and find it. (b) Show that the limit doesn't exist finitely if we replace $\sqrt2$ by $1.5$. What are ...
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Infinite Power Tower approximation: float error? [closed]

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - ...
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Does $\Im(e^i+e^{e^i}+e^{e^i+e^{e^i}}\dots)$ converge?

Consider the following sum (where $\Im(z)$ denotes the imaginary part of $z$) $$\Im(e^i+e^{e^i}+e^{e^i+e^{e^i}}\dots)$$ I.e; $$\Im(\lim_{n\to\infty}a_n)$$ $$a_1=e^i,\ \ \ a_{n+1}=a_n+e^{a_n}\ \ \ \...
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60 views

Euler's paper 'De formulis exponentialibus replicatis' in English

Is there an English version of this paper of Euler's: De formulis exponentialibus replicatis? I can not find one on Euler Archive or by internet search.
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Uniqueness criterion for tetration based on signs of derivatives?

Consider the fractional iterations of the expontential function denoted $$ \exp^{[t]}(x) $$ $$ \exp^{[0]}(x) = x $$ $$ \exp^{[t]}(x) = \exp(\exp^{[t-1]}(x) $$ $$ \exp^{[l + m]} = \exp^{[l]}(\exp^{[m]}(...
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how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$

Use $y=x^{\frac{1}{x}}$ graph and think the following calculate. $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$$ I want to know how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$ will be. When $\sqrt{2}^{\...
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35 views

Tetration as a basis?

Is there a way to use tetration with powers of 2 as a unique additive basis that's analogous to the binary representation (i.e. 100101)?
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1answer
86 views

How to solve $x^x-x=1$?

I was recently posed the question "solve for $x$ in $x^x-x=1$". The intended answer was $x=0$, assuming that $0^0=1$, but I used brute force and determined another solution, $x\approx1....
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How do I write a super-logarithm in a graphing application such as Desmos?

I'm trying to plug in a super-logarithm (inverse function of tetration) in Desmos Wikipedia's definition though is defined implicitly: https://en.wikipedia.org/wiki/Super-logarithm Not sure how to ...
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Does $^{\infty}i$ actually converge to $\frac{2i}{\pi}W(-\frac{\pi i}{2})$?

The question Using only the principle branch of the complex log. Skipping to the end, if $z = \frac{2i}{\pi}W(-\frac{\pi i}{2})$, which is the fixed point of $z \rightarrow i^z$, it's not true that a ...
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“Shooting Room” ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
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Is there a way to calculate the zeros of $f(z,w)= w-z^{(z^w)}$?

I know the zeros of $f(z,w)=w-z^w$ have an analytic form: $$\operatorname{zero}[z,n]=-\frac{W[-\log(z),n]}{\log(z)}$$ Is there a way to compute the zeros of $$f(z,w)=w-z^{(z^w)}$$?
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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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380 views

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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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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2answers
63 views

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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49 views

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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138 views

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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156 views

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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62 views

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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1answer
109 views

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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