Questions tagged [power-towers]

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

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Using matrix algebra, can we show that the infinite power series [(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1[? [closed]

Can the answer be related to the assumption :[For positive symmetric physical power matrices, the sum of their eigenvalues is equal to the eigenvalue of their sum of power series]
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Digits tower power iterate

Stack the digits of a natural number into a power tower, iterate until only one digit remains. Does this iteration always terminate for any positive integer? Additionally specify $0^n = 0$, even when $...
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Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$

Hi I want to show the following fact : Problem : Let $x\geq 1$ and $n\geq 1$ a natural number and define: $$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$ Then we have : $$f''(x)\ge 0$$ ...
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2 votes
1 answer
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$f(n)=$, for even integer’s $n$, $\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$

$f(n)=$, for even integer’s $n$, $$\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$$ $^n x$ is the tetrative function or ‘power tower’, which when $n=\infty$ can also be written in terms of y, ...
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44 votes
4 answers
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Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

A few years ago I asked about the inequality Prove that $\int_0^\infty\frac1{x^x}\, dx<2$. As I came back to revisit it, I found that each of the following tetration integrals $$\int_0^\infty\frac{...
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Examples of closed forms of integrals with a power tower argument using W-Lambert function.

Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ...
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1 vote
2 answers
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Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$

One can learn that for a power tower of height $n$: $$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$ giving a recursive relation. One might see that $n<-2$ cases are ...
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4 votes
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Could someone explain why $\sum_{\substack{a_1,\ldots,a_n\in\mathbb{N}_0\\a_1+\cdots+a_n=n}}\frac{n!}{a_1!\cdots a_n!}=n^n$?

I want to use this equality but I have no idea why it holds. Sure I can probably prove it via induction but it looks rather fiddly. (Let $n$ be a positive integer.) $$\sum_{\substack{a_1,\ldots,a_n\in\...
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Is anything significant known about inverting power towers?

This has to be something somebody has looked into, but I can't find it. In general, for finite expressions of the form $${a_1}^{{{a_2}^{{⋰}^{a_i}}}},$$ if one reverses the usual order of ...
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2 answers
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About the inequality $f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$

Prove or disprove that $-1< x<0$ then we have : $$f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$$ Where : $$f(x)=2-x^{-\frac{212}{1000}x^{x^{\frac{1}{5}x^{2x^{\frac{1}{5}x}}}}}$$ My ...
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7 votes
1 answer
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Can $\int\limits_0^\infty e^{ix^ x}dx$ be written without a limit?

We know about the Fresnel Integrals: $$C(x)=\int \cos x^2 \, dx,\quad S(x)=\int \sin x^2 \, dx$$ which can also be written as: $$\int e^{ix^2}dx=C(x)+i\,S(x)$$ To make a more interesting and tetration ...
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8 votes
1 answer
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Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using ...
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2 votes
0 answers
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Is there any formula to sum the series $\sum_{k=1}^n{2^{k^k}}$

$$\sum_{k=1}^n{2^{k^k}}$$ I couldn't find anything on this with a simple Google search so is it not trivial? If yes then can someone link some resources/Wikipedia page where I could read more? (can't ...
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Does $\mathrm{\int W(ln(x))dx}$ have a closed form?

This is follow up to this question which you will have to see for context: Is there a better solution for $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(...
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2 answers
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Solution to the Equation $x^x = 0$.

If I solve the Equation $x^x = 0$, then I get $x^x=0$: x ln⁡(x)= -∞ or ln⁡x $e^{\ln⁡(x)}$ =-∞ and if we take the Lambert W function on both sides we get ln⁡(x)=∞, And I put it on Wolfram Alpha then it ...
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4 votes
1 answer
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Prove or disprove that the function is convex .

It seems we have : Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$ Claim : $f''(x)\geq 0$ My attempt : The case $n=1$ is trivial . So I have ...
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4 votes
2 answers
493 views

Correct my proof about : $x^{x^{x^{x^{x^{x}}}}}$ is convex for $0.25<x<0.27$

Let $0.25<x<0.27$ then define : $$f(x)=x^{x^{x^{x^{x^{x}}}}}$$ Claim: The second derivative of $f(x)$ is strictly positive . Proof : Let $0.25<x<0.27$ then define : $$g(x)=x^{x^{x^{x}}}$$ ...
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5 votes
1 answer
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Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{\,ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\ln(a))}{n^{n+1}}\,dt}$?

I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are ...
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4 votes
1 answer
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How to evaluate the finite power tower $\tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$

Consider the following finite power tower: $$\Large \tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$$ I'm wondering if there is a way to solve this that doesn't rely on ...
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6 votes
1 answer
161 views

Is the infinite product $\prod_{i=0}^{\infty}(1+\frac{1}{2^{3^i}})$ transcendental?

Is the following number algebraic or transcendental? $$P:=\prod_{i=0}^{\infty}\left(1+\frac{1}{2^{3^i}}\right)$$ We could also define it as follows: let A be the set of natural numbers which contain ...
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1 vote
1 answer
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On $\int \sqrt[x]x$dx. Solution found for $eW\left(\frac1e\right)\le x\le e$.

Notice: Note that there is hope for a more general answer because an answer for the area under the infinite tetration/power tower curve was almost certainly found. It uses the nice OEIS A008405 find. ...
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15 votes
1 answer
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About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
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3 votes
0 answers
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Stronger statement : $^{6}x\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$

Let $x>0$ then prove or disprove that : $$x^{x^{x^{x^{x^{x}}}}}\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$$ My attempt : ...
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17 votes
2 answers
796 views

Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.

I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one ...
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19 votes
3 answers
737 views

Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$

Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have : Claim : $$f''(x)\geq 0$$ My attempt as a sketch of partial proof : We introduce the function ($0<a<1$): $$g(x)=x^{x^{a^{a}}}$$ Second ...
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7 votes
2 answers
529 views

Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?

A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real ...
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7 votes
1 answer
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Limits of negative-power tower

Consider the following: $$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$ It doesn't converge for $m \to \infty$, but ...
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2 votes
1 answer
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If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
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1 vote
1 answer
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did I make a mistake? when trying to find the derivative of $x^{x^{x^{x^{x^{\dots}}}}}$

did I make a mistake? when trying to find the derivative of $f(x)=x^{x^{x^{x^{x^{\dots}}}}}$ the first thing I did was to change the $f(x)$ to this $$y=x^y$$ I took the derivative of both $y$ and $x$ ...
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0 votes
0 answers
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Representing a Number as the Sum of Powers of Form $k^k$.

So I was wondering if it would be useful to instead of writing a number in base $2$ or $3$, we use functions in general as bases. So like writing it as the sum of squares or other increasing functions....
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3 votes
0 answers
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Solution Verification - Integral of infinite power tower

$$\int x^{x^{\cdot^{\cdot^\cdot}}}dx=\int -\frac{W(-\ln(x))}{\ln(x)}dx$$ Let $u=-\frac{W(-\ln(x))}{\ln(x)}$, $\frac{du}{dx}=\frac{u^2}{(1-\ln(u))u^{\frac{1}{u}}}\Rightarrow dx=\frac{(1-\ln(u))u^{\frac{...
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7 votes
1 answer
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Find a value of $\;\lim\limits_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$

Find a value of $$\lim_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$$ For your information$,\quad\uparrow\uparrow$ is a tetration defined as $$a\uparrow\uparrow n:=\...
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2 votes
2 answers
165 views

Integrating an Infinite exponent tower

$$ \int_0^1 x^{2^{x^{2^{x^{\ldots}}}}} ~~ dx = ~~?$$                                                                                                                                           What I'...
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0 votes
6 answers
161 views

Is it correct for $\int {x^x}\, dx $?

Recently, I was working with power towers. I was interested in $x^x$; it is easy to know its derivative, but I wanted to find its integral. Here's what I found. Please let me know if it is right or ...
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3 votes
1 answer
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Area under $\frac{1}{x^x}$ curve

How can I calculate area under $\frac{1}{x^x}$ on any interval, I tried the Archimedes method, but I get $$\frac1n\sum \frac 1{X_n^{X_n} }$$ and that's very complex to calculate because of the roots, ...
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3 votes
1 answer
81 views

Solution verification: Derivative of the infinite power tower $y(x) = x^{x^⋰}$

I was doing the problem $(x^{x^{⋰}})'$ and I would like someone to verify my solution: \begin{align*} &\left(y=x^{x^{^{⋰}}}\right)'\\ \implies & \;\left(y=x^{y}\right)'\\ \implies & \;\...
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1 vote
1 answer
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What is 333^333^333 in mod 17

Is there an easier way to do this than finding cycles of different mods? Or can I just first do 333 (mod 17), it gives me 10. Then I could change all the 333's into 10s so it would be 10^10^10 and ...
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1 vote
1 answer
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Solving $a_{n+1} = x^{a_n}$ for various $x$ [duplicate]

(a) Consider a sequence defined by $a_0=1$ and $a_{n+1}=(\sqrt2)^{a_n}$ . Prove that limit exists and find it. (b) Show that the limit doesn't exist finitely if we replace $\sqrt2$ by $1.5$. What are ...
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2 votes
1 answer
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Exponential Power Tower

My question is- $$(4x)^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.0625$$ How to solve it? Options- (A)$2^{1/24}$ (B)$2^{1/48}$ (C)$4^{1/48}$ (D)$2^{1/96}$ I am confused how to solve this infinite ...
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-1 votes
1 answer
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Infinite Power Tower approximation: float error? [closed]

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - ...
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5 votes
2 answers
130 views

Comparing power towers of $2$s and $3s$

Let $x=[x_1,x_2,...,x_n]$ be a finite list of positive real numbers, and define $\tau x$ as the power tower formed by these numbers. The function $\tau$ can be recursively defined by the following two ...
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1 vote
0 answers
22 views

inequality $\Re\Bigg(\Bigg(1-\frac{W(-\ln(2a))}{\ln(2a)}\Bigg)^{1-\frac{1}{a}}+\Bigg(1-\frac{W(-\ln(2b))}{\ln(2b)}\Bigg)^{1-\frac{1}{b}}\Bigg)<1$

Problem inspired by Vasile Cirtoaje : Let $0<b<0.5<a<1$ such that $a+b=1$ then we have : $$\Re\Bigg(\Bigg(1-\frac{W(-\ln(2a))}{\ln(2a)}\Bigg)^{1-\frac{1}{a}}+\Bigg(1-\frac{W(-\ln(2b))}{\...
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0 votes
2 answers
95 views

how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$

Use $y=x^{\frac{1}{x}}$ graph and think the following calculate. $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$$ I want to know how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$ will be. When $\sqrt{2}^{\...
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4 votes
1 answer
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Is it possible to express $\int_{0}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0}^{1}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ function seems is not ...
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1 vote
0 answers
79 views

"Shooting Room" ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
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0 votes
1 answer
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Does the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converge for $x < e^{-e}$?

As per the answer at https://math.stackexchange.com/a/573040/23890, the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converges if and only if $ x \in [e^{-e}, e^\frac{1}{e} ] $. Is $...
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1 vote
1 answer
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$i^i\in\mathbb R$

I knew that $^2i=i^i\in\mathbb R$ so I tried to find another number $n\in\mathbb N$ like that: $^ni\in\mathbb R$, so, there is no such number all the way up to a million, can we prove that two is the ...
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0 votes
1 answer
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Why it were conjectured that $e^{e^{^e{^{79}}}}$ is not an integer only for $n=79$ ? any non trivial characterization?

I'm confused that why exactly and what is the reason to conjecture that $e^{e^{^e{^{79}}}}$ is not integer , why not for example with $n=87$ or any other prime $p$ ?Is this number special ? or is ...
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12 votes
2 answers
385 views

Convergence of probabilistic power tower $e^{\pm e^{\pm e^{...}}}$

While pondering over this question, I came across another interesting one. I am familiar with infinite tetration and its convergence over the reals. Nevertheless, when I saw this power tower, I couldn'...
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1 vote
0 answers
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Almost integer with nested radicals and power tower .

playing with power tower and nested radicals I get : Prove that Let $a_1=\sqrt{2}$ ,$a_2=\sqrt{2}^{\sqrt{2}}$,$a_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}$,$a_4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}...
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