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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How to calculate generalized Puiseux series?
I recently posted this, post containing a series of questions concerning the integration of ${x^{x^{x^x}}}$. In order to do so, I wrote ${x^{x^{x^x}}}$ as the following infinite summation:
$$1 + \sum_{...
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Questions concerning the Integration of Integer Tetration
I've been interested in finding the antiderivative of integer tetration, a function defined as iterative exponentiation. Integer tetration is written as $^n$$x$ where $^1$$x =x$, $^2$$x =x^x$, $^3$$x =...
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Sum of a series and tetration?
Is there an equation that can represent the value of the sum of a series (a sigma) if a tetration takes place inside it?
For example:
$$
\sum\limits_{K=1}\limits^{N} {}^2\!K.
$$
(Here ${}^2\!K = K^K$....
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another number group?
I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
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4answers
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Solve $x^{x^x}=-1$
I do know that if you use tetration the equation would look like this.
$$^3x=-1$$
You could then theoretically use the super-root function to solve the equation, but I do not know how to use the ...
4
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1answer
90 views
Can different tetrations have the same value?
Suppose, we have two numbers
$a\uparrow \uparrow b$ and $c\uparrow \uparrow d$.
To avoid trivial cases, suppose $a,b,c,d>2$ and $(a,b)\ne (c,d)$.
Is there a quartupel $(a,b,c,d)$ with $a\...
2
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1answer
55 views
Doubt in raising a power to a complex number [duplicate]
What's the value of $$i^{i^{i^{...}}}$$?
I tried to take log on both sides.
$x=i^x$
$\implies \log x=x \log i$
After this how can I solve this...
I am sorry, that I don't know the methods you ...
2
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3answers
206 views
Will $a^a$ ever out-grow $9^{9^{^\ldots}}$?
I am trying to come up with the largest finite number that can be made using a set number of characters.
I have two expressions which are calculated and printed out by a program (theoretically - they ...
5
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1answer
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Solving for $a$ in power tower equation
$$n=a^{(a+1)^{(a+2)^{(a+3)\cdots}}}$$
How would one go about solving in this equation? I am more used to solving equations in this form:
$$n=a^{a^{a^{a\cdots}}}$$
Which you solve in this form:
$$a^...
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0answers
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Super root function [closed]
Super root is an invertion of tetration. Lets define $f(x) = \sqrt[x]{x}_s$. Definition makes sense when x is integer. Is there an extension of this function to real numbers? Similar how Gamma ...
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Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?
The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
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3answers
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Is this a valid proof for ${x^{x^{x^{x^{x^{\dots}}}}}} = y$?
So I got this challenge from my teacher.
Solve ${x^{x^{x^{x^{x^{\dots}}}}}} = y$ (eq. 1) for $x$.
My attempt:
As $x^{y^z}$ per definition equals $x^{y \cdot z}$, then $x^y = y$ from (eq. 1). Thus, $...
2
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1answer
398 views
Limit involving tetration
Let the notation be $a^{\wedge\wedge}b
= \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration.
My mentor conjectured the following:
Let $n$ be a positive integer, then let $A(n)$ be ...
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2answers
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How to prove that $x\uparrow \uparrow 1/2 = \sqrt x_s$ [closed]
This may be a stupid question but when we work with exponentiation we can see that $x^{\frac 12}=\sqrt x$ because:
$x^{\frac 12}\times x^{\frac 12}=x^{\frac 12+\frac 12}=x^1=x$
and
$\sqrt x \times \...
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1answer
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The Physical Meaning of Tetration with fractional power tower
I've read a passage in the forum about tetration and did some research on Wiki. I understand the basic definition for any real height $n>-2$,
$$^na=a^{a^{a^{a}}}...\text{, for real height =n}$$
...
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Continuum between addition, multiplication and exponentiation?
I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one.
However, I was looking for the more general case where we find ...
9
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1answer
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A problem in understanding infinite towers (tetration)
To solve equations involving power towers (infinite tetration) we usually do something like this:
$$x^{x^{x^{x^{\dots}}}} =k$$
$$x^{(x^{x^{x^{\dots}}})} =k$$
$$x^k=k$$
$$x=\sqrt[k]k$$
But what if ...
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3answers
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Algorithm for tetration to work with floating point numbers
So far, I've figured out an algorithm for tetration that works.
However, although the variable a can be floating or integer, unfortunately, the variable ...
0
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1answer
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Inverse and named fixed values, with ↑↑?
The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log.
Continuing upwards hyperoperationally, what is the inverse of $↑↑$?
Whats more somtimes values that are fixed are given names, ...
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1answer
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Last Digits of a Tetration
I was studying tetrations, or "power towers", and I found a decently well-known fact. The last $k-1$ digits of $^k 3 = 3^{3^{\vdots^{3}}} (k \text{ threes)}$ remain constant, for all numbers $^a 3$ ...
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1answer
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Graham's number of layer 1 tetration explanation?
I have a question on how the number of the first layer of the Graham's number ($g_1$) is computed.
From Wikipedia:
http://en.wikipedia.org/wiki/Graham%27s_number#Magnitude
$g_1 = 3\uparrow\uparrow\...
4
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1answer
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For what values does $x^{x^{x^{x^{.^{.^{.}}}}}}$ make sense [duplicate]
For which values of $x\ge 1$ does the expression $x^{x^{x^{x^{.^{.^{.}}}}}}$ make sense?
To tackle this, define $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$ for $x \ge 1$ and $n\ge1$.
a) Show that $f_{n+...
2
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1answer
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What number tetrated by itself equals a googol?
http://en.wikipedia.org/wiki/Tetration
Tetrating stuff makes it really big really fast. I'm trying to figure out what number would equal a googol. Any help?
Or a googolplex ?
7
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1answer
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Growth of $n!!\dots !$
The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as
$$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$
Is there a similar formula for the iterated ...
9
votes
2answers
441 views
Do the last digits of exponential towers really converge to a fixed sequence?
While fooling around with exponential towers I noticed something odd:
$$ 3^{3} \equiv 2\underline{7} \mod 100000 $$
$$ 3^{3^{3}} \equiv 849\underline{87} \mod 100000 $$
$$ 3^{3^{3^{3}}} \equiv 39\...
2
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0answers
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How to differentiate $(x!)\uparrow\uparrow(!x)$?
I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials...
$$(x!)\uparrow\uparrow(!x)$...
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Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?
There are the following common notations:
Sums:
$$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$
Products:
$$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$
Is there a (theoretical) one for:
Exponentiation (...
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1answer
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How to prove that a tetration holds the same last digit after a certain rank?
Let's say $u_{n} = 7^{u_{n-1}}$ and $u_{0} = 7$, how can we prove that for $n > 0$, that we will get the same last digit?
I know that the last digit of $7^7$ is 3, we can prove it using modulo/...
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1answer
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Which one is greater?
For any $x\in\mathbb{R}^+$, let $x\diamond 1=x$ and $x\diamond (n+1) = x^{x\diamond n}$ for $n\in\mathbb{N}$. For example, $2\diamond 3 = 2^{2^2}=16$.
If $t$ be an unique positive real number such ...
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3answers
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How can I solve this equation $x^{x^{x^{x^{.^{.^{.}}}}}}-a=0$
I always use the Newton-Raphson Method if I want to find the roots of any equation as follow
$$x_{1}=x_{0}-\frac{y_{0}}{y'_{0}}$$
But I don't know how to use this method if the equation takes the ...
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4answers
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Finding solutions to $ x^x = 2x$
A friend claims it isn't possible to find a closed form for the smaller positive real solution of $x^x = 2x$. Numerically we have seen that $0.346...$ and $2$ are solutions, but are failing to do ...
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1answer
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complex and decimal tetration
So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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2answers
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A peculiar observation about infinity.
Let ${\sqrt2^\sqrt2}^{\sqrt2^...}=y$.
Then $\sqrt 2^y=y$
$\implies \sqrt 2=y^{1/y}$
$\implies \sqrt 2 =1$
$\implies 2 =1$ !! but how come that be. Can anyone explain this and point out what is ...
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3answers
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How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]
How can I prove
$$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$
I don't know which method can be used for this?
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0answers
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Which primes satisfy this modular property?
Let $x$ be a residue$\mod p$ where $p$ is an odd prime.
Im searching for such $p$ such that there exists a function $f(x)$ with propery
$f(f(x)) - 2^x \equiv 0 \mod p $
for all values of $x$.
I ...
4
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0answers
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Tetrations of non-integers? [duplicate]
A Tetration is defined as $${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$$ or, by a recursion function, $${^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{...
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2answers
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Is there an inverse operation of tetration for values between 0 and .3?
So I don't understand everything with tetration, but the graph of $^2x$ or $x^x$ does not have any values between 0 and .3 on the y axis. So if we were to trying inverse tetration on say .2 what is ...
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1answer
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How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?
In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
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3answers
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What combinatorial quantity the tetration of two natural numbers represents?
Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. $$\...
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Explain $x^{x^{x^{{\cdots}}}} = \,\,3$
$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$
This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...
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2answers
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Is there a closed form for the inverse of $y=x^{x^x}$?
It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
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1answer
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Iteration of $x \to x^x$
If $u(x)=x^x$ then we can form
$$
u^2(x) = \left(x^x \right)^{x^x} = x^{x^{x+1}}
$$
some simplification occurs, but the further iterates are a typographical challenge to mathjax.
Writing $E_k$ for $...
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1answer
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Convergence of an infinite power
There are complex numbers $z$ and $w$ for which
$$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$
where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$.
Are there complex ...
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3answers
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How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]
Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$
Is it possible to find $f'(x)$. If yes, please show all steps.
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1answer
104 views
Is it true that ${^\infty}{\sqrt[x]{x}} = x$
I was fiddling around with tetration and I stumbled across an interesting idea, ${^\infty}{\sqrt[x]{x}}$. I messed around with the concept a little bit and I had the following idea:
Let ${^\infty}y = ...
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1answer
112 views
After Exponentials
Look at this-
$$f(a,b)=a+b$$
The next step would be to make a function $g$ such that $$g(a,b)=\underbrace{a + a + a \cdots}_{b\text{ times}}=a\cdot b$$
Then we made $h$ so that $$h(a,b)=\underbrace{a\...
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2answers
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If a uniquness for all functions exist shouldn't there be uniquness to recursion?
What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example.
Let's make a table of values for $^{x}2$
(0,1)
(1,2)
(2,4)
(3,...
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Clarification on tetration
So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$
For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
2
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2answers
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Tetration graph for this function
I'm trying to visualize the derivatives of exponential tetration, by taking the original equation from its graph. Right now there is no elementary way of expressing the derivative. I'm not allowed to ...
2
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1answer
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A question on the equation $^qx=2$
Given the equation $$^qx=2$$ with $q\gt3$ where $^qx$ means the 'tetration' operation on $x$, my question is: is it possible to find a value for $q$ for which the solution $x$ of the equation is a ...
