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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Solving an equation with infinite exponents [on hold]

How would I find the exact value of this infinite power tower? $$(1+2^{0})^{(1+2^{-1})^{(1+2^{-2})^{.^{.^{.}}}}}$$ I have found a decimal expansion of the number through calculation of the first few ...
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35 views

Tetration of a number giving a complex number

Giving this power equation: $$S=\lim_{n\to\infty} {^n}x=-i$$ where the symbol $^nx$ means the tetration operator, we can write in a form not formally correct: $${\ ^{n}x = \ \atop {\ }} ...
3
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3answers
85 views

The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
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77 views

Equality of power towers : $a\uparrow\uparrow m=b\uparrow \uparrow n$

Suppose, $a,m,b,n$ are natural numbers greater than $1$. If we have $$a\uparrow\uparrow m=b\uparrow\uparrow n$$ can we conclude $a=b$ and $m=n$ ? $a\uparrow \uparrow m$ is a powertower of $m$ $a's$ ...
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48 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 ...
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1answer
28 views

What is a puiseux series and what is wolfram-alpha doing with this antiderivative?

I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the ...
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3answers
355 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
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0answers
40 views

Last three digits of tetration

Find the last three digits of the number: $7^{7^{7^7...}}$ where there are 1001 sevens. I know how to do it for when there are 4 and 5 sevens. I get an answer of 343. But how do I find it for ...
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0answers
45 views

Sum of the reciprocal of tetration?

Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth. What are f(2) and f(3), and could you please also explain how you reached these answers?
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1answer
54 views

Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence

Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. ...
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44 views

Showing that a Hermitian matrix can have eigenvalues that correspond to arbitrary numbers does not prove the Hilbert-Polya conjecture, does it?

I read in Wikipedia about the Hilbert-Polya conjecture that: " ...a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts ...
3
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2answers
67 views

How fast does this sequence grow?

I have the following recursive definition of a sequence of numbers: $$a_{n+1}=(a_n)^{(a_{n-1})}$$ And $a_0=a_1=2$. The first few terms are: $$a_2=4$$ $$a_3=16$$ $$a_4=65536$$ $$a_5=1.1579209 ...
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45 views

Derivation of tetration by iteration

I was screwing around a bit differentiating tetrations and was trying to write some rules for them. I came up with this recursive definition: $$ \frac{d\ ^nt}{dt} =\ ^nt \cdot log(t)\frac{d\ ...
3
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2answers
374 views

Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^...}}}$ does not converge, that is ...
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1answer
96 views

Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
2
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1answer
66 views

How would you define non-integer tetration? [duplicate]

Tetration is defined for all $n\in \Bbb{N}$ by $$ {^1}a = a \\ {^{n+1}}a = a^{\left({^n}a\right)} $$ Thus ${^3}a$ means $a^{a^a}$. Here $a$ could be any real (or indeed even complex) value, but only ...
2
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2answers
126 views

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham's number and before we can even start calculating G1 which is $3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of $3 ...
2
votes
2answers
356 views

Convergence or divergence of infinite power towers of complex numbers $z^{z^{z^{z{…}}}}$

Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence ...
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1answer
111 views

How do I write Grahams number

I found that graham's number is :enter image description here So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?
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1answer
46 views

Question concerning comparison of different tetration functions

Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$ Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$ Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$? If so, is the proof elementary? (Use only ...
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2answers
219 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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29 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 = ...
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4answers
482 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 ...
3
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3answers
122 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 ...
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1answer
89 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 ...
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3answers
90 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 ...
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2answers
159 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
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4answers
289 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 ...
2
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1answer
106 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 ...
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1answer
91 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 ...
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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} ...
6
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4answers
234 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 $\sqrt[n]y_s$ ...
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1answer
117 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
7
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104 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 ...
0
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0answers
41 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
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1answer
164 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
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2answers
64 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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1answer
65 views

Calculate this power? [closed]

Let $a$ is a positive real number. If ${a^{{a^{{a^{16}}}}}} = 16$ how much is ${a^{{a^{{a^{12}}}}}}$?
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2answers
71 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: $$ ...
5
votes
5answers
297 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 ...
0
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1answer
59 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 ...
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votes
2answers
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
votes
3answers
147 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
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0answers
121 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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1answer
89 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
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0answers
52 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 ...
11
votes
4answers
483 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 ...
3
votes
1answer
84 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
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 ...
2
votes
3answers
187 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 ...