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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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Is tetration and quintation etc. of infinite cardinals well defined?

I would like to know whether tetration and quintation functions are well defined for infinite cardinals, thus, for example, $$\aleph_0 \text{ [tet] } \aleph_0 = \aleph_0 ^ {\aleph_0 ^ {\aleph_0^{\...
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Tetration with 0 < #s < 1?

When I try numbers between 0 and 1 on my calculator app n-calc on my phone .. I get rapid convergence when the numbers are close to 1 .. And alternating but slow convergence when numbers are close to ...
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function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$

Here it's cited: the existence of the holomorphic function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$ had been demonstrated in 1950 by Hellmuth Kneser. ...
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$i_a = (i_b + i_c + 1)/2$ and $f(x) = f(x - f(x-1)) /2$?

Consider the set $i_0,i_1,...$ defined as $i_0$ is the smallest element and $i_0 = 0$. If $ (i_a - i_b)^2 < 1$ then $i_c$ is Also in the set and given by $$ i_c = (i_a + i_b + 1)/2$$ [*] Let $T(...
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Series of the form $\sum_{n=1}^\infty \frac{1}{ ^{s}n}$ where $^{s}n$ is the tetration of $n$

I am interested in the function $$T(s)=\sum_{n=1}^\infty \frac{1}{ ^{s}n}$$ where $s$ is an integer and $ ^s n$ is $n^{n^{n^{...}}}$ $s$ times. I know these series are convergent since tetration of $n&...
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Can tetration 'escape' the complex plane?

Considering subtraction can break out of the natural numbers and into integers, division can break out of integers and into rational numbers, and exponentiation can break out of rational numbers and ...
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351 views

Solutions of $a^{a^x}=x$ for fixed $a>0$

I am interested in the equation $a^{a^x}=x$ for some fixed $a>0$. Is there some way to rearrange for $x$ or solve otherwise? What about the nature of the solutions? For which fixed $a>0$ are ...
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160 views

Addition, Multiplication, Exponentiation, Tetration. Iterated functions.

Please note that I am exceptionally talented in over-complicating even the simplest of topics, however this may be worth a read. Also, the following operations are only performed with positive ...
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82 views

How to compute the upper (repelling) fixpoint for $b^x=x$ ($1<b<1.44$) - using the LambertW?

For the computation of the lower (attracting) fixpoint $L_0$ for $b^{L_0} = L_0 $ (where the base $b$ is in the range for convergence $1 \lt b \lt e^{1/e}$) there is the simple formula $$L_0 = \exp(-W(...
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$f ^{\prime \prime}(x) = f(x) f^{\prime \prime} (x-1) $

I know the equation $$ f^{\prime} (x) = f(x) f^{\prime} (x-1) $$ is solved by $f(x) = C$ or by tetration ( $ f(x+1) = \exp(f(x)) $). So I wonder What are the solutions to $$g^{\prime \prime}(x) =...
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Variation of Fermat number, looking for literature on special properties

Suppose we define a recursive function as follows: $F_k(0) = 1$ For $i \epsilon \mathbb{N}, i>0$, we define: $F_k(i) = k^{F_k(i-1)}$ So if $k=2$ we get: $F_2(3) = 2^{2^2}$ $F_2(4) = 2^{{2^2}^...
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Pentation Notation - How does it work? [duplicate]

When going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^3}}=...
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Analytically continuing the series exponentially exponential series?

Background I was recently toying with a series: $$ S = \exp(x)+ \exp(\exp(x))+ \exp(\exp(\exp(x)))+ \dots $$ Taking exponential both sides: $$ \implies e^S = \exp(\exp(x)) \cdot \exp(\exp(\exp(x)))...
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“Derivative-like” Operator such that $H(f(x)^{g(x)}) = H(f(x))^{H(g(x))}$

Given 2 functions $f(x),g(x)$ we have: $$D(f(x)+g(x)) = D(f(x))+D(g(x))$$ And $$D^*(f(x)g(x)) = D^*(f(x))D^*(g(x))$$ where $D^*$ is the multiplicative derivative: $$D^*(f(x)) = lim_{h\rightarrow 0} {\...
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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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1answer
121 views

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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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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2answers
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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
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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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A new interesting pattern to $i↑↑n$ that looks cool (and $z↑↑x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-...
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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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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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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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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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2answers
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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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6answers
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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
155 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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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
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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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2answers
50 views

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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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
594 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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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
42 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
310 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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300 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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0answers
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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
142 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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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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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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82 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
171 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, ...
4
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0answers
185 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 ...
6
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1answer
51 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 ...
2
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1answer
76 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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149 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 ...