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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Solve $(x+a)^{1/x} = b$ for $x$ [on hold]

Solve $(x+a)^{1/x} = b$ , for $x$ where $a$ & $b$ are real constant. note that this equation can be also written as: $e^{2ni\pi}(x+a)^{1/x} = b$ when dealing with powers but consider that $n=...
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44 views

separate $a$ & $b$ in $ssrt(a^a*b^b)$

It is already known that $ssrt(a^a*b^b)$ does not equal $ssrt(a^a)*ssrt(b^b) = a*b$ Is there any other method to separate $a$ and $b$? ****Please note that $ssrt$ is "super square root". and my ...
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49 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
12
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1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
1
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1answer
103 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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36 views

question concerning tetration to infinity.

So I read on-line that \begin{equation} x^{x^{x^{x^{x^{x}}}}}=2 \end{equation} where number of x goes to infinity can be solved by solving \begin{equation} x^{2}=2 \end{equation} so by the same logic \...
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1answer
66 views

If $|t| = |W(-\ln z)| = 1$ and $t^n =1$ then $z^{z^{z^{…}}}$ is convergent

Let $z \in \mathbb{C}$ and $W$ be the Lambert W function. In this post I was told if $|t| = |W(-\ln z)| = 1$ and $t^n =1$ for some $n \in \mathbb{N}$ than the iterated exponential $z^{z^{z^{...}}}$ ...
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62 views

Solve $x*(x^x)=a$ for $x$

I tried with Lambert W-function but it seems it can not solve it. $x*x^x = a$ $a$ is constant find $x=f(a)$
2
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121 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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3answers
138 views

How to prove that $x\uparrow \uparrow 1/2 = \sqrt x_s$

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 \...
14
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522 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
2
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1answer
56 views

How can I show, that $N\uparrow\uparrow N$ is not “much larger” than $N$ for very large $N\ $?

Here : https://sites.google.com/site/largenumbers/home/3-2/knuth Saibian demonstrates that for very large numbers $N$, $N\uparrow\uparrow N$ is only "slightly larger" than $N$. I would like to ...
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1answer
112 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp\left(\frac{-K \cdot (m - a(n))}{m}\right),\ n \geq 1$?

Edit: In the original post, I put the function $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ which is not the function I wanted to study. The correct one is the one given below I came up ...
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508 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 $n \geq 1$, ...
2
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2answers
149 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 \uparrow\uparrow\...
3
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3answers
109 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 $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
2
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36 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 {\ }} {{\underbrace{...
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1answer
95 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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2answers
133 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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70 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
30 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 ...
2
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42 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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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=...
3
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2answers
419 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 $|W(-\...
2
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2answers
371 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 $z$...
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48 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
55 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. ...
2
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46 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\ ^{n-1}t}{...
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48 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
73 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 \...
3
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1answer
114 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
118 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?
2
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1answer
67 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 ...
14
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2k views

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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2answers
161 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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1answer
49 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 ...
2
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2answers
225 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?
2
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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 ...
19
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4answers
491 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 ...
6
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240 views

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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31 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 = \sqrt{2}^{\sqrt{2}^{\...
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347 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 ...
3
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3answers
124 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 D'...
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1answer
93 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 simplification:...
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465 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
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4answers
294 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
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 ...
1
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1answer
144 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\...
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
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2answers
436 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?