# 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.

318 questions
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### How to ompute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $\int{x↑↑n} dx$ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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### Has something like “Knuth's Up-Arrow Factorial Notation” ever been used? If so, what practical uses does it have?

I was studying Knuth's up-arrow notation and I was wondering if ever something like "Knuth's up-arrow factorial notation" has ever been used. Now I know this probably isn't a recognizable term, ...
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### A new interesting pattern to $i\uparrow\uparrow n$ that looks cool (and $z\uparrow\uparrow x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to ...
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### Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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### How to solve $x^x = y^y \mod p$?

Let $p>5$ be a prime. How to solve $x^x = y^y \mod p$? How many solutions are there for a given $p$ such that $x,y < p$? I know the discrete logarithm and the theory of quadratic residues, but ...
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### How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
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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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### 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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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\... 4answers 155 views ### 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 ... 1answer 85 views ### Exponent is to exponentiation as _______ is to tetration Would it be tetrand, tetrant, or something else? 0answers 195 views ### Prime factors of$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$ The two smallest are$6771419$and$72153167$, which I found by trial division. The number ... 2answers 82 views ### 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} =... 0answers 31 views ### 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 ... 0answers 47 views ### 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 ... 0answers 23 views ### 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 ???? 2answers 34 views ### 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 ... 2answers 296 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 ... 2answers 109 views ### 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 ... 1answer 145 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 ... 0answers 42 views ### 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)$$In general ^{k}... 0answers 26 views ### Maximum of exponential tower First I introduce a notation similar to \sum_{i=1}^n a_i for exponentiation. I.e. for any (potentially infinite) sequence a_i we define$$ ES_{i=1}^n a_i = \left\{\begin{matrix} a_1 &... 1answer 135 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^...
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 ...