If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...

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How can i count number of digits in tetrated numbers?

As you read in the title, I need a technique for counting number of digits in tetrated numbers. For example: ${3^{(3)}}^3 = 7625597484987 $(13 digits) ${7^{(7)}}^7 = $How many digits (...
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139 views

Exponentiation and far too high numbers?

I love very, very, very, big numbers! You see, I'm working on powers of $2$ and I need to calculate the next expression in this sequence: $2^2=4$ $2\uparrow\uparrow2=216$ $2\uparrow\uparrow\uparrow2=...
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160 views

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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127 views

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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93 views

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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53 views

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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489 views

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 ...
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85 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\...
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48 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 ...
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3answers
191 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 ...
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167 views

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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141 views

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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336 views

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
177 views

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,...
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333 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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145 views

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
187 views

The Physical Meaning of Tetration with fractional power tower

I've read a passage in the forum about tetration and had do some research on Wiki. I understand the basic defination for any real height n>-2, $$^na=a^{a^{a^{a}}}...\text{, for real height =n}$$ I ...
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561 views

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
314 views

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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466 views

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 ...
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68 views

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 ...
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138 views

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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155 views

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\...
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131 views

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+...
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1answer
188 views

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 ?
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144 views

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 ...
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206 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\...
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0answers
123 views

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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112 views

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
49 views

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
61 views

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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214 views

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
209 views

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
108 views

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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134 views

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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413 views

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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41 views

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 ...
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84 views

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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84 views

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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147 views

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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457 views

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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1k views

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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156 views

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
106 views

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
62 views

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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399 views

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
92 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
104 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
75 views

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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255 views

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 $1$ ...