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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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2answers
190 views

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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For which $a\in\Bbb R^+$ is $\{1,a, a^a, a^{a^a},…\}$ linearly independent over $\Bbb Z$?

Can I choose a positive real number $a\in\Bbb R^+$ so that $1,a,a^a,a^{a^a},...$ are independent in the sense that no combination of integer coefficients will add up these numbers to zero? More ...
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1answer
203 views

Tighter bounds on the fast growing hierarchy?

Not a dupe of this question, as I'm searching for tighter bounds. We define the fast growing hierarchy for finite values as follows: $$f_k(n)=\begin{cases}n+1,&k=0\\f_{k-1}^n(n),&k>0\end{...
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143 views

The solution of $x^x=2$ rational/algebraic irrational/transcendental?

What does the unique real number $x$ such that $x^x=2$ equal to? Is the value rational, algebraic irrational or transcendental? What about $x^x=3$? Or $x^x=e$? $x^x=π$?
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Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
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1answer
368 views

Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?

For a long time (a couple of years) I'm following the Q&A's about "half-iterate of $\exp(x)$" etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder's method, but also a $\mathbb R \...
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1answer
53 views

Partial iteration and inverse iteration of $e^x$

Find the general $f^n(x)$ where $f^1(x)=e^x$ $f^{a}(f^{b}x)=f^{(a+b)}(x)$ where $n,x\in\mathbb R$ I'm fairly confident that no two $f^n(x)$ with different values of $n$ will intersect, ...
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3answers
1k views

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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222 views

How to extend this extension of tetration? [closed]

if $0\le b<1$, then $a↑↑b = a^b$ if $b\ge1$, then $a↑↑b = a^{a↑↑(b-1)}$ if $b<0$, then $a↑↑b = \log_a(a↑↑(b+1))$ so for example, $2↑↑\pi = 2^{2^{2^{2^{0.1415926...}}}} = 21.5963561$ How can ...
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98 views

Is this a good reasoning on fractional tetration?

Searching stuff on Wikipedia somehow I got to tetration, and got really interested on how could an interpretation of fractional tetration be given. So I did the following analysis $$ \left(a \cdot \...
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2answers
85 views

How many ways can a sequence of $1$s be partitioned into pairs or singles?

How many distinct ways can a sequence of $n$ $1$s be partitioned into pairs or singles, in which $\{1,1\}=\{2\}$ is considered a pair and $\{1\}$ is considered a single? For example $\{1,1,1,1\}$ can ...
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332 views

Infinite Power Tower

I've been having fun with the problem of finding the values of $n$ for which the infinite power tower $$\sqrt{2}^{\sqrt{2}^{...^{\sqrt{2}^n}}}$$ Has a finite value. My final answer was that it ...
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How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\...
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How do you calculate $ 2^{2^{2^{2^{2}}}} $?

From information I have gathered online, this should be equivalent to $2^{16}$ but when I punch the numbers into this large number calculator, the number comes out to be over a thousand digits. Is the ...
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1answer
211 views

Derivative of super square root [closed]

What is the derivative of $y=^{1/2}x$? I tried finding the derivative of $x^{x}$ and then finding the inverse of that, but that didn't work.
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1answer
164 views

Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations?

Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the ...
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1answer
236 views

Can any number of factorials overtake the tetration function ${^x}10$?

Tetration is repeated exponentiation evaluated from right to left. The value of both the factorial and tetration function at $x=0$ is defined to be 1. So,both functions start at $x=0$ (when the values ...
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How can I define a function that raises a number to the power of itself a given number of times?

We can use the following to add the number $2$ to itself $5$ times. $$f(n,k) = \sum_{x=1}^k n = n\cdot k$$ $$2 + 2 + 2 + 2 + 2 = f(2,5) = \sum_{x=1}^5 2 = 2\cdot 5 = 10$$ We can use a similar ...
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Weird result regarding “infinitely explosive” differential equations

Firstly, take the family of differential equations $\dot x = > \frac{dx}{dt}=x^\alpha$, for any $\alpha \in \mathbb R$ The solution to these equations is $$(\text{for } \alpha=1):x(t)=x_0e^t$$ $$(\...
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Prove that if $a=\,^xx$, for $x>2$, where $2\,|\,x$ and $10\nmid x$, the last digit of $a$ is $6$.

First, explanation of some notations. $^wv$ is called tetration, which is a higher order of exponentiation. Useful link -> https://en.wikipedia.org/wiki/Tetration. $v\,|\,w$ means "$v$ divides $w$", ...
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141 views

Is this a correct way to extend the definition of super-logarithms?

I think I may have found something new because it's giving the correct results. I'm using the functional-roots of $logx$ to calculate super-logarithms. You can read this post of mine to get the idea : ...
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5answers
835 views

Half iteration of exponential function

I'm working on the half iteration of the exponential function. No one has any idea what fractional iterations could mean but I think intuitively it should be a function $f(x)$ such that $f(f(x))=e^x$. ...
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156 views

How to find a function $f(x)$ such that $f(f(x))=\log_ax$?

Is there some method to do this or maybe some method to find a function such that $f(f(x))$ is at least approximately equal to $\log_{a}x$? Maybe the taylor series could be of help. So, we're looking ...
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About the smallest fixpoint of $exp(qz)$.

I got inspired by this http://math.eretrandre.org/tetrationforum/showthread.php?tid=1149 Where Tommy assumes problems for tetration. I was intrested in a closed form for when the fixpoint is on the ...
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1answer
55 views

General form of self-composed function

Given some $f(x)$ composed with itself $n$ times, how would one go about finding a closed-form expression in terms of $x$ and $n?$ Specifically, I'm trying to find a function in two natural number ...
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2answers
399 views

Why is tetration considered the next step after exponentiation? [closed]

Tetration is often stated to be the next step after exponentiation (see for example Wikipedia): $$\large a^{a^{a^{...^a}}}$$ Where the exponents are taken $b$ times from the top. I refer to the ...
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638 views

What is the value of $^{\frac{1}{10}}e$?

By $^nx$, I mean $x$ tetrated to $n$. So, basically, I'm looking for the solution of the equation $$\large x^{x^{x^{x^{x^{x^{x^{x{^{x^x}}}}}}}}}=e$$. Is there some way to find the approximate value by ...
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37 views

What types of fractional tetrations are possible to calculate without using extensions of tetration?

For example, any number tetrated to a positive integer can be calculated by just doing repeated exponentiation from top to bottom. Similarly, it is possible to calculate values of $^{0.5}2$, $^{3}3$, $...
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1answer
169 views

Show that a certain parametric equation does not represent a cardioid

For $z \in \mathbb{C}$ consider the sequence $z, z^z, z^{z^z} ... ,$ that is, the iterated exponential with base $z$. If $z$ belongs to the Shell-Thron region the iterated exponential will converge. ...
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2answers
409 views

A limit related to asymptotic growth of tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
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1answer
144 views

Attempt on fractional tetration

To start off, I was looking at the following ingeniously made form of the Gamma function: $$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$ which lies on the back of $$1=\...
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86 views

Fatou Coordinate with two complex conjugate fixed points and extending Tetration to real values

Lets us define $f(z)=z^2+z+c$ with real valued c>0 and iterate the function f(z). Then the Abel function for $f(z)$ is $$\alpha(z)\;\; \text{where}\;\; \alpha(f(z))=\alpha(z)+1$$ $$ f^{[\circ z]} = \...
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2answers
183 views

Finding the value of n, so that it is bigger than M?

We introduce some notation for writing really big (but finite) numbers. A googol, denoted g, is defined by $g = 10^{100}$. A googolplex, denoted G, is defined by $G = 10^g$. A MathPatharoo, denoted M, ...
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291 views

When does $x^{x^{x^{…^x}}}$ diverge but $x^{x^{x^{…^c}}}$ converge?

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $c\ne x^c$ $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ to converge under ...
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1answer
52 views

Find a sequence such that this tower of of exponent is convergent

Context We already know that if we take a sequence $(x_n)\in{\mathbb R_+^*}^{\mathbb N}$ such that $$x_n=O\left(\frac 1{n^2}\right)$$ then $$\sum_{n=0}^\infty x_n <+\infty.$$ We also now that ...
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1answer
79 views

Asymptotic to $f( 2 f^{[-1]}(x) ) $? [closed]

Let $f(x)$ be the half-iterate of $ 2 sinh(x).$ Im looking for an asymptotic to $f( 2 f^{[-1]}(x))$ for Large $x>0$. $^{[*]}$ means iteration here thus $^{[-1]}$ means functional inverse. For the ...
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157 views

Peter Walker's $C^{\infty}$ conjectured nowhere analytic slog

Consider the function $h(x)$ which is conjectured to be $c^\infty$ nowhere analytic, and can be used to generate what we call the base change slog, which is the inverse of the basechange sexp. This ...
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3answers
143 views

Is $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}$ rational, algebraic irrational, or transcendental?

Let $$ x=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}. $$ Assume $x$ is algebraic irrational. By the Gelfond-Schneider Theorem, $x^x$, which is also $x$, is transcendental, a contradiction. But I ...
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2answers
500 views

Complete monotonicity of a sequence related to tetration

Let $\Delta$ denote the forward difference operator on a sequence: $$\Delta s_n = s_{n+1} - s_n,$$ and $\Delta^m$ denote the forward difference of the order $m$: $$\Delta^0 s_n = s_n, \quad \Delta^{m+...
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1answer
189 views

A limit related to tetration growth rate

Assume $a\in\mathbb R,\,e^{-e}<a<e^{1/e}$ and $n\in\mathbb N$. Let ${^n a}$ denote tetration: $${^0a}=1,\quad{^{(n+1)}a}=a^{\left({^n a}\right)}.\tag1$$ It is well known that under these ...
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1answer
101 views

Is there a mathematical term, practical application, or area of math that covers a function raised to itself?

Some abstract examples would be: $f(x)^{f(x)}$ or $f(x)^{f(x)^{f(x)...}}$ Actual equations I've attempted to look at can be viewed here on desmos.com There seems to be a pattern of common convergence, ...
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2answers
171 views

Solving for best fit value $C$ in $\sqrt {Exp_a^{[1/2]} (x) \cdot Exp_b^{[1/2]} (x )}$ ~~ $ Exp_C^{[1/2]} (x).$

Let $Exp_t^{[y]} (x) $ denote the $y$ th iteration of the exponential function with base $t$ : $t^x.$ For example $Exp_t^{[1]} (x) = t^x. $ Let ~~ denote best fit. Now as $x$ Goes to positive ...
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1answer
173 views

What is the value of $i^{i^{i^\ldots}}$? [closed]

What is the value of $i^{i^{i^\ldots}}$? My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg ...
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2answers
641 views

How to solve this tetration equation $\;^n 2 = \;^2 n $?

How would one find all real solutions to the following equation: $\qquad$ $n^n = 2^{2^{2^{2^{\dots^2}}}} $(where the number of $2$s is equal to $n$) generalizing to $n$ being a real value. In ...
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0answers
100 views

Generalizing an iterative logarithm integral.

First, some notation: $f^{\star0}(x)=x$ $f^{\star k}(x)=f\left(f^{\star k-1}(x)\right)$ So that for any integer $k$, $\log^{\star k}(x)=\underbrace{\log(\log(\dots(\log}_k(x))\dots))$. I then came ...
5
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2answers
183 views

How fast do iterated exponentiation converge?

Iterated exponentiation is defined by $$x \mapsto x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$$ or more conveniently, we denote by $^rx$ the expression $\underbrace{x^{x^{\cdot^{\cdot^{\cdot^{x}}}}}}_{r \text{ ...
6
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1answer
97 views

Can we use trigonometric idendities to calculate $\cos(x)$ and $\sin(x)$ for extremely large $x$?

If we want to calculate $\sin(x)$ and $\cos(x)$ for very large $x$ , lets say $10^5$ , the usual way is to reduce the number $x$ modulo $2\pi$. If the number is a large power of a small number, for ...
5
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2answers
765 views

What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?

In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but ...
1
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1answer
171 views

The one value of super square root function

A function can not have more than one value. i.e. we only take the positive value for $y$ where $y=\sqrt 4$ But what about the super square root?, if both value are positive real values like: $$y=\...
3
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1answer
214 views

Prove if $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ than $z^{z^{z^{…}}}$ does not converge

Let $z \in \mathbb{C}$ and let $W$ be the Lambert W function. I am trying to prove that: If $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ for all $n \in \mathbb{N}$ (ie, $t$ is not a root of unity) ...