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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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126 views

Convergence of the sequence $x_{n+1} = a^{x_n}$ [duplicate]

Let $a > 0$. Show that the sequence defined by $$ x_0 = 1, \qquad x_{n+1} = a^{x_n} $$ converges for $a \leq e^{1/e}$. Any help is appreciated, I don't even know where to start with this. Edit to ...
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Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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Nested Tetration properties

Regarding tetration, I know properties like ${}^a({}^bn)= {}^{ab}n$ do not hold in general. When $a=b=2$, for instance, we have $$ {}^2({}^2n)={}^2\left(n^n \right)=\left(n^n \right)^{\left(n^n \right)...
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Finding the derivative of $x$ tetrated to the $x$

Differentiating the functions $x^x$, $x^{x^x}$ (or ${^2{x}}$ and ${^3{x}}$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this ...
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Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$

I'm a computer student, learning math just for fun. Today I was graphing for fun that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to ...
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Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
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1answer
55 views

The factors of a tetration plus an integer

There I was, just messing around with tetration, when I stumbled across this - $(x^x +1)/(x+1)$ = integer (for odd integer values of x) Playing some more with this it seems (not entirely sure as ...
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1answer
85 views

Prove that ${}^ni$ is complex for all $n \ge 3$

We can define ${}^nx$ as $\underbrace{\displaystyle {x^{x^{\cdot ^{\cdot ^{x}}}}}}_{n\text{ times}}$ (Tetration). I conjecture that ${}^ni$ is complex for all $n \ge 3, n \in \mathbb{N}$. I've ...
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41 views

inverse tetrations of complex numbers

for the complex function $f_{(2)}(z)=z^z$,where in the complex plane does the inverse $z_{(2)}(f)$ not exist, same for inverse of functions $f_{(3)}(z)=z^{z^z}$ being $z_{(3)}(f)$ and so on for ...
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1answer
125 views

Uniquely extended fractional iterations of $\exp$

Let us define the following basic conditions for an iterated exponential function: $$\exp^1(x)=e^x\tag{$\forall x$}$$ $$\exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$}$$ I then pondered what ...
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Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges?

let $f_1(x)=f(x)$ $f_2(x)=f(f(x))$ $f_3(x)=f(f(f(x)))$ and so on... Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges? It is okay if the ...
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thoughts about $f(f(x))=e^x$

I was thinking, inspired by mathlinks, precisely from this post, if there exists a continuous real function $f:\mathbb R\to\mathbb R$ such that $$f(f(x))=e^x.$$ However I have not still been able to ...
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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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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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Interpolation between iterations of exponentials (and logarithms)

I'm interested in finding a continuous and ideally smooth family of real-valued monotonic functions $H_t(x)$ that interpolate between iterations of exponentials, and also iterations of logarithms. So ...
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$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
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1answer
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Compare power towers

Prove or disprove: $3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours. Taking logs is useless, and there seems to be no other way to compare. Thanks!
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Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, … don't change, and generalisation

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
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Find all ordered triples of primes $(a, b, c)$ such that $a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$.

Find all ordered triples of primes $(a, b, c)$ such that $$\large a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$$ Notation: $$\large {^xy} \text{ or } x^{\underline y}$$ are nonations for ...
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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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When raising to a power and tetrating, which comes first? Is $^24^3$ equal to $(^24)^3$, or to $^2(4^3)$?

I know tetration isn't quite an used operation, but anyway, what if it were featured in an expression? For example, what does $^24^3$ mean? Is it $(^24)^3=(4^4)^3=4^{12}$ or is it $^2(4^3)=^2{64}=64^{...
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How iterated exponential $\exp^{[\circ x]}(y)$, $y\neq 1$, defined based on tetration?

Background: The tetration \begin{equation} ^xe = \exp^{[\circ x]}(1) = \underbrace{e^{e^{\cdot^{\cdot^e}}}}_{x \text{ times}} \end{equation} is well defined when $x \in \mathbb{Z}$. The extension of ...
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Proof (or hints towards proof) for asymptotic shape of orbit $0 \to 1 \to b \to b^b \to \cdots$ with certain class of $b$?

In a paper by Baker & Rippon (1983) the property of being convergent or divergent for iterated exponentials $z_{h+1} \to b^{z_h}$ with $b$ complex and $z_0=1, z_1=b, z_2=b^b, \cdots$ for classes ...
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Calculating the nth super-root when n is greater than 2?

Tetration (literally "4th operator iteration") is iterated exponents, much like how exponents are iterated multiplying. For example, $2$^^$3$ is the same as $2^{2^2}$, which is $4^2$ or 16. This ...
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Find range of $a$ if $\{x_n\}$ converges where $x_{n+1}=a^{x_{n}},n=1,2,\ldots$ and $x_1=a$ [duplicate]

For $a>1, x_{n+1}=a^{x_{n}}, n=1,2, \ldots$ and $x_{1}=a$. If $\{x_{n}\}$ converges, find range of $a$. $\begin{aligned} x_{n+1}-x_{n}&=f\left(x_{n}\right)-f\left(x_{n-1}\right)\\ &=f^{\...
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1answer
286 views

What is the maximum of $\sum_{k=1}^{\infty} (-1)^k(^kx)$?

During my testing of the series $\sum\limits_{k=1}^{n} (-1)^k(^kx)$, I found that the sum converges to two limits when $n \to \infty$, for $e^{-e} \lt x \le e^{1/e}$ and oscillates between depending ...
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Proving the derivation of the infinite tetration via Summation/Product notation is equivalent to the derivation via Implicit Differentiation

First some background info. Feel free to skip to the bold text below if you already know this. I) The derivative of the infinite tetration of $x$ (I guess that's how you say it) or ${}^\infty x$ ...
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Tetration of cos : An original Inequality .

It's a little bit crazy for me the problem I propose to you today : Let $a,b,c\geq 0$ such that $a+b+c=1$ then we have : $$3\cos(\frac{1}{3})^{\cos(\frac{1}{3})^{\cos(\frac{1}{3})}}\geq\cos(a)^{\...
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1answer
135 views

What Tetration Algorithm can I utilize

I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow": A mandelbrot fractal (I call it "operation order 1") is done by ...
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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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Tetration of $\pi$: Can it be a prime number?

Given the tetration as \begin{align} {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n \end{align} and the set of prime numbers as $\mathbb{P}$. Can you prove or to disprove the following ...
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Quickly show that $\Bigg|(-1/2)^{(-1/2)^{(-1/2)}}\Bigg|=e^{\pi\sqrt{2}}$

Question: Is it true and can we quickly show that $$\Bigg|(-1/2)^{(-1/2)^{(-1/2)}}\Bigg|=e^{\pi\sqrt{2}}$$ Here was my solution. Abusively I write $a/b$ for the fraction ${a \above 1.5pt b} $. I ...
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$w =\operatorname{arcsinh}(1+2\operatorname{arcsinh}(1+2^2\operatorname{arcsinh}(1+2^{2^2}\operatorname{arcsinh}(1+\dotsm$

In the context of positive reals consider$\DeclareMathOperator{\arcsinh}{arcsinh}$ $$ w= \arcsinh( 1 + 2 \arcsinh( 1 + 2^2 \arcsinh ( 1 + 2^{2^2} \arcsinh( 1 + 2^{2^{2^2}} \arcsinh( 1 + \dotsm $$ ...
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3answers
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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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1answer
54 views

Is there a “natural tetration function”?

For the natural exponential function, $$ f(x)=e^x \to f(x)=f'(x). $$ Is there a natural tetration (tetral?) function? $$ f(x)={{^x}b} \to f(x)=f'(x) $$ Is it base $e$?
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How to compute 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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1answer
101 views

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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443 views

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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1answer
61 views

Continuous, infinitely differentiable tetration map

We know that the exponential map is infinitely differentiable; is there a tetration map $TET$ defined on $(0,\infty)$ such that 1) $TET(x) > TET(y)$ when $x > y$ 2) $TET(e^x) = TET(x) +1$ 3) $...
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1answer
25 views

Power tower(infinite tetration) [duplicate]

How to find the domain and range of $Y=x^{x^{x^{x^{x^{...}}}}}=x^Y$ I know how to differentiate the function. I don't know how to proceed further. We should prove that domain:$[1/(e^e),e^{(1/e)}]$ ...
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1answer
55 views

Pattern in power towers of 2 involving last digits

We have \begin{align} 2^{2^{2}} &\mod 10 = 6 \\ 2^{2^{2^2}} &\mod 100 = 36 \\ 2^{2^{2^{2^2}}} &\mod 1000 = 736 \\ 2^{2^{2^{2^{2^{2}}}}} &\mod 10000 = 8736 \\ 2^{2^{2^{2^{2^{2^2}}}}} &...
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Associocommutativity

One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring ...
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3answers
827 views

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 ...
8
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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=\...
3
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2answers
330 views

Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
3
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1answer
436 views

Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than $...
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5answers
690 views

Fractional Composite of Functions

I would like to know how I can calculate a fractional composition of a function. Let be $f(x)$, where $x \in R$ and $f(x) \in R$. I now how to do $f(f(x))=f^2(x)$. Now suppose I would like to do $f^{\...
14
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5answers
1k views

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 ...