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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Negative Tetrations?

To start, I'll say that for this post I'll be using Rudy Rucker notation for tetration. That being $^2$x=$x^x%$, which means the number raised to the left means how many times one would exponentiate x....
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General formula for the derivative of a pentation?

Now, many people probably know of the first three hyperoperations, such as addition, multiplication, and exponentiation. However, many don't know of the fourth, tetration, and even less then know ...
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Can we characterize functions, like exponentials, the gamma function, and tetration, as solutions of an optimization problem?

This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to ...
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General Rule for Differentiation of Tetrations

I'll start at the beginning. Initially, this sort of began as just what is $\frac{d}{dx}$[$x^x$] the answer being $x^x$+ln(x)$x^x$. This wasn't difficult to achieve, just some chain rule and product ...
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What is the equivalent of the factorial function, but for exponentiation?

The factorial function multiplies a given number by each number less than itself until reaching one. Does a function, notation, or literature yet exist regarding the idea of raising a given number to ...
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Does anyone know if it's possible to solve $x^x=x+1$ in terms of $x$?

So I have tried solving for $x$ algebraicly using the productlog function but all I was able to do is: $$x\log(x) = W(x\log(x)(x+1))$$ Maybe I could use the square-super root formula $e^{W(\log(x))}$, ...
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Is field of complex numbers closed under tetration?

The set of real numbers is closed under multiplication, but not under exponentiation (Eg. square root of negative numbers). That is, $\exists a, b \in R \mid {a^b} \notin R$. Then we introduced ...
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Can we generalize the tetration to the real or complex numbers? [duplicate]

is it possible to find a value for this operation: ${^{(3/2)}2}$? If so, can we generalize the domain of the function ${^{x}a}$ to the real or even complex numbers? I had originally tried to solve the ...
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Is there a proof showing super roots and super logarithms won't lead to a solution for the quintic?

So I am learning about tetrations and I just learned that tetrations are not elementary functions. When I heard that I remembered back to the statement that there is no general solution to the quintic ...
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Hyperoperators and zero-value conditions

Define for complex numbers: $f_0(z) = z+e; f_{n+1}(z+1) = f_n(f_{n+1}(z))$, where e = 2.1828... can be replaced with other bases. It's not hard to see that this is satisfied by: $f_1(z) = ez; f_2(z) = ...
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Examples of closed forms of integrals with a power tower argument using W-Lambert function.

Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ...
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Why is $\sum_{n\ge1}\frac{\text B_\frac n2(-n,n)}{n^a b^n }=-\sum_{n\ge1}\frac{\left(\frac 2n-1\right)^n}{n^{a+1}b^n}$ close to (reciprocal) integers?

Here is a possible closed form of a sum with tetration in it made by equating coefficients of the Incomplete Beta function. This question is inspired by: Closed form of $$\sum\limits_{n=1}^\infty \...
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Did I prove $\;{^{n}2} \equiv {^{n-1}2} $ $\,$ mod $n\;\;\;$ for $\;n \geq 2\;$?

Let's first state some properties of tetrations: They are one of the basic arithmetic functions, 4. hyperoperation to be exact. $\;{^{n}2}$ is in its basic, most elemental form. So we can't really ...
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Find all integers $n, n\gt2$ such that $n^{n-2}=x^n$ for some $x$

We can express this alternatively as $n^{n}=n^{2}x^{n}$. So the number raised to the power of itself has to be proportional to some number to the $n$-th power, but cannot be equal, naturally. I am not ...
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Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$

One can learn that for a power tower of height $n$: $$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$ giving a recursive relation. One might see that $n<-2$ cases are ...
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Why are tetrations not useful? [closed]

I've always wondered after learning addition, multiplication, and power facts (and their inverse operations) what the next higher level of facts I would need to memorize would be. However, instead of ...
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Could someone explain why $\sum_{\substack{a_1,\ldots,a_n\in\mathbb{N}_0\\a_1+\cdots+a_n=n}}\frac{n!}{a_1!\cdots a_n!}=n^n$?

I want to use this equality but I have no idea why it holds. Sure I can probably prove it via induction but it looks rather fiddly. (Let $n$ be a positive integer.) $$\sum_{\substack{a_1,\ldots,a_n\in\...
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Is there a way to write the $n^\text{th}$ super root in terms of the lambert W function?

Super root is one of the inverse functions of tetration defined as- $y=^nz$ $\implies z=\sqrt[n]{y}_s$ We can easily get an infinite series representation of $\sqrt[n]{z}_s$ using the Lagrange ...
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Question about complex number infinite tetrations

I was researching infinite tetrations recently and thought of a super interesting, although admittedly pointless, math question regarding it. I was wondering if anyone can help me solve it. Say we ...
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Closure under reverse tetration

The natural numbers are closed under addition, but not subtraction The integers are closed under multiplication, but not division The rationals are closed under exponentiation, but not roots The real ...
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Can $\int\limits_0^\infty e^{ix^ x}dx$ be written without a limit?

We know about the Fresnel Integrals: $$C(x)=\int \cos x^2 \, dx,\quad S(x)=\int \sin x^2 \, dx$$ which can also be written as: $$\int e^{ix^2}dx=C(x)+i\,S(x)$$ To make a more interesting and tetration ...
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Can a differential equation result in a tetrational growth?

For an ordinary differential equation $$f(y^{(n)}(x), y^{(n-1)}(x),...,y''(x),y'(x),y(x),x) = 0$$ where $f$ can be written in a closed form using only elementary functions, is it possible for its ...
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Repeated Exponentiation doubt [duplicate]

I have a somewhat silly doubt about this seemingly basic topic - repeated exponentiation. I've got very little experience regarding this, so please hear me out. We know that one of the properties of ...
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8 votes
1 answer
317 views

Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using ...
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7 votes
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Does $\mathrm{\int W(ln(x))dx}$ have a closed form?

This is follow up to this question which you will have to see for context: Is there a better solution for $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(...
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Solution to the Equation $x^x = 0$.

If I solve the Equation $x^x = 0$, then I get $x^x=0$: x ln⁡(x)= -∞ or ln⁡x $e^{\ln⁡(x)}$ =-∞ and if we take the Lambert W function on both sides we get ln⁡(x)=∞, And I put it on Wolfram Alpha then it ...
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2 votes
1 answer
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Is $\exp(a\exp(a\exp(a\cdots)))$, where $a=\pi/2$, a valid representation of $i$?

Take $i=e^{\frac{i\pi}{2}}$. For this question it will be more convenient to write it as $i=e^{\frac{\pi}{2}i}$. Substituting in this value for $i$, we get $$i=e^{\frac{\pi}{2}e^{\frac{\pi}{2}i}}$$ ...
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435 views

Prove or disprove that the function is convex .

It seems we have : Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$ Claim : $f''(x)\geq 0$ My attempt : The case $n=1$ is trivial . So I have ...
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6 votes
1 answer
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Solve $\sum_{n=1}^\inftyΓ(-n,-n)\mathop= \pi\left(\frac1e-1\right)i+ \sum_{n=1}^\infty \frac{(-1)^n \text{Ei}(n)}{n!}+\sum_{n=1}^\infty a_n・(-e)^n $

Motivation: It may be possible to find an integral representation using: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+...
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5 votes
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Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{\,ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\ln(a))}{n^{n+1}}\,dt}$?

I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are ...
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4 votes
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How to evaluate the finite power tower $\tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$

Consider the following finite power tower: $$\Large \tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$$ I'm wondering if there is a way to solve this that doesn't rely on ...
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1 vote
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On $\int \sqrt[x]x$dx. Solution found for $eW\left(\frac1e\right)\le x\le e$.

Notice: Note that there is hope for a more general answer because an answer for the area under the infinite tetration/power tower curve was almost certainly found. It uses the nice OEIS A008405 find. ...
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3 votes
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Categorification of tetration, is it possible?

Following this question How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory? I was asking myself if a weaker tetration-like object can ...
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3 votes
1 answer
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Alternate non-sum form or integral representation of $\sum_{x\in \Bbb Z} (-1)^x \left(x^\frac1x -1\right)$.

I was inspired from this problem with an integral of an infinite tetration. I saw that you can do this series for the infinite tetration with an analytic continuation. However, the Here is the sum ...
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17 votes
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Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.

I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one ...
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1 vote
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Square rooting in tetration.

I have been interested in tetration for a while and I have this question: what is the tetration-root of $2$? ($x^x=2$). In tetration form, it can be written as 2^^x. In general, I am just curious on ...
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Infinite sum of powerseries likely converges to a powerseries with rational coefficients and has then a simple generating function... proof?

Background: In a discussion in the "tetration-forum" a term $\log(x-L)/L+\log(x-L^*)/L^*$ occured, where $L$ and $L^*$ mean the (complex) primary fixpoint (and its conjugate) of the $\exp()$-...
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2 votes
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Proving $\forall d\in\mathbb{Z_+},\ \min\{n:\lambda^n(10^d)=1\}=d+2,$ where $\lambda()$ is the Carmichael function.

(I'm posting this question in the spirit of this advice and will post an answer if no one else does so.) [This paper] proves the following: If $k, x$ and $a_1,a_2,a_3\dots$ are positive integers, ...
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7 votes
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Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?

A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real ...
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Clarify a step in proving $\forall n\ge 1,$ the sequence $2, 2^2, 2^{2^2}, 2^{2^{2^2}}, \dots$ is eventually constant $\pmod n$. [duplicate]

The 1991 USAMO Problem 3 offers a concise proof-by-contradiction that if $n$ is any positive integer, then the sequence $2, 2^2, 2^{2^2}, 2^{2^{2^2}}, \dots$ is eventually constant $\pmod n$. However, ...
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can a number be written as another number to its own power

Do we know which numbers can be written as another number to its own power? More precisely, if we have a number $x$, when can we write it as $n^n$ for some $n\in \mathbb{N}$? For example, if $q$ is a ...
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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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2 votes
1 answer
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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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1 vote
0 answers
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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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5 votes
2 answers
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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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0 votes
2 answers
104 views

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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25 votes
2 answers
621 views

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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