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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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 ...
15
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3answers
180 views

An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?

Start with $i=\sqrt{-1}$. This will be $a_1$. $a_2$ will be $i^i$. $a_3$ will be $i^{i^{i}}$. $\vdots$ etc. In Knuth up-arrow notation: $$a_n=i\uparrow\uparrow n$$ And, amazingly, you can ...
0
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0answers
15 views

What the minimum of infinite tetration divided by $\sqrt{x}$?

For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point $$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = ...
5
votes
4answers
323 views

Tetration limit

For each $n$, define $f_n:\mathbb R^+\rightarrow \mathbb R^+$ by $f_n(x) = \underbrace{x^{x^{x^{...^{x^x}}}}}_n$ Is it true that $\lim\limits_{n \to \infty} f_n(\frac{n+1}{n}) = 1$ ? A few ...
2
votes
3answers
114 views

Infinite exponentials

We can read a lot of about convergence of series or Infinite products. E.g. for series. Following series $$\sum_{i=1}^\infty a_i$$ is convergent when $$\lim_{n\rightarrow\infty}a_n=0$$ and ...
0
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1answer
70 views

Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after ...
6
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3answers
431 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 ...
3
votes
4answers
227 views

Mathematical fallacy of $x^{x^{x^{x^x…}}}$ = 2

Suppose we have an equation with an infinite number of $x$'s as an exponent: $$x^{x^{x^{x^x...}}} = 2$$ $$x^{(x^{x^{x^x...}})} = 2$$ because there are infinity $x$'s in the parentheses, which we've ...
0
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3answers
88 views

Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
1
vote
1answer
123 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 = ...
21
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3answers
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 ...
2
votes
2answers
402 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
6
votes
1answer
136 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 ...
1
vote
1answer
62 views

Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
5
votes
5answers
197 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 ...
0
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1answer
111 views

Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
3
votes
2answers
143 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 ...
66
votes
4answers
4k views

Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = ...
25
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2answers
2k views

Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
1
vote
1answer
110 views

How do I evaluate this summation?

I was wondering how do I solve the summation? $$\frac{1}{n}\sum_{j=1}^nja_{m,j-1}a_{m-1,j-1}$$ I found it in this link on power towers http://mathworld.wolfram.com/PowerTower.html
6
votes
4answers
225 views

A limit related to super-root (tetration inverse).

Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ ...
1
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1answer
84 views

Very confused about a limit.

This question is about where I made my mistake in the computation of a limit. It relates to An answer I gave that confused me. The question to which I gave the (partial) answer is related to ...
3
votes
1answer
134 views

Tetration Binomial Theorem

I was exploring tetration and came across the following identities: $${^0}(ab) = 1$$ $${^2}(ab) = ({^2}a)^b * ({^2}b)^a$$ $${^3}(ab) = (ab)^({^2}(ab)) = (ab)^{(({^2}a)^b * ({^2}b)^a)}$$ That third ...
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0answers
29 views

Is $\max \int_{- \infty}^{\infty} \frac{dx}{\pi (1 + x^2 + f ' (x)^2 )} $ a uniqueness condition here?

Let f(x) be a real-differentiable function with $f′(x)>0,f′′(x)>0 $ and $$ f(f(x)) = \exp(x) $$ for all real $x$. Tommy1729 adds the optimization condition $$ max \int_{- \infty}^{\infty} ...
7
votes
0answers
91 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 ...
11
votes
4answers
472 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 ...
9
votes
1answer
146 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
0
votes
0answers
32 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' ...
0
votes
1answer
94 views

Infinite tetration convergence

I came across infinite tetrations on wikipedia (https://en.wikipedia.org/wiki/Tetration) it says that the infinite tetration converges if and only if $\ e^{-e} \leq x \leq e^{1/e}$. I was wondering if ...
5
votes
2answers
54 views

Finding the Final Digits in a Stacked Exponents Problem

How to find the the last two digits of the number $$\underbrace{\huge 7^{7^{7^{...}}}}_{1+n}$$ When there are $n$ sevens in the ascending exponents. I have no idea how to condense this or ...
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votes
1answer
65 views

Calculate this power? [closed]

Let $a$ is a positive real number. If ${a^{{a^{{a^{16}}}}}} = 16$ how much is ${a^{{a^{{a^{12}}}}}}$?
5
votes
2answers
69 views

Checking primality for $2 \uparrow \uparrow n + 3 \uparrow \uparrow n$

Is there a clever way to test primality for $$2 \uparrow \uparrow n \quad + \quad 3 \uparrow \uparrow n$$ where $n \gt 3$? (Not surprisingly, I got stuck after that) For $n \le 3$ we get: $$ ...
4
votes
5answers
279 views

Finding ${i^i}^{i\cdots}$

We know that surprisingly enough, $i^i=\frac1{e^{\frac\pi2}}$. But what about finding the value of ${i^i}^{i\cdots}$? Is it possible? My attempt: Let $${i^i}^{i\cdots}=x$$ $$i^x=x$$ Or ...
2
votes
1answer
108 views

Is the exact value of $^{\infty}i$ known? [duplicate]

Wikipedia's article on tetration has a table of successive tetrations of $i$ that seems to imply that $^{\infty}i$ converges, and my own experimentation seems to confirm it, but I'm suspicious of ...
21
votes
8answers
735 views

Infinite powering by $i$ [duplicate]

Find the value of: $i^{i^{i^{i^{i^{i^{....\infty}}}}}}$ Simply infinite powering by i's and the limiting value. Thank you for the help.
11
votes
2answers
627 views

Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
0
votes
1answer
52 views

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 ...
-1
votes
1answer
115 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
votes
3answers
139 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 + ...
2
votes
0answers
105 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 ...
1
vote
1answer
77 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 = ...
2
votes
0answers
50 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 ...
3
votes
1answer
83 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 ...
2
votes
1answer
44 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 ...
0
votes
1answer
136 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 ...
11
votes
1answer
162 views

Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
5
votes
1answer
152 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: ...
13
votes
2answers
280 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, ...
4
votes
0answers
110 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 ...
1
vote
1answer
325 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 ...