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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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Summation of :$\sum_{n=2}^{\infty} \frac{nx^n}{\ln(n)} $ [closed]

Can anyone help me to compute this Sum: $$\sum_{n=2}^{\infty} \frac{nx^n}{\ln(n)} $$ ; thank you..
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Solve $x^{x^{x^\ldots}}=n^n$ for $x$.

Solve $x^{x^{x^\ldots}}=n^n$ for $x$. Let $x^{x^{x^\ldots}}$ be an infinite power tower. Then we can fairly quickly solve $x^{x^{x^\ldots}}=n$ to find the solution $x=n^{1/n}$ But what about for $x^...
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1answer
28 views

Calculating accumulation

I would firstly like to point out that I am just becoming very interested in maths but have very little technical mathematical knowledge, so I, therefore, understand that the phenomenon that I'm ...
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0answers
55 views

Is there a simpler way to sum the first $n$ terms of the sequence of numbers starting at 2 and squaring to get the next?

I've got this summation: $$ f(n)=\sum_{i=0}^{n-1}2^{2^i} $$ In effect, it's the sum of the sequence of numbers you get from starting at 2 and squaring the previous number in the sequence: $$ \begin{...
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4answers
60 views

Equation whose solution is a finite tower of $2's$

What equation has a finite power tower of $2's,$ as it's solution: $$ x=2^{{2^{2}}^{\cdot\cdot\cdot}}.$$ I tried to reverse engineer the solution, back into an equation, so I started with a toy ...
3
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2answers
72 views

Compare different base powers-towers (of 'height' five)?

Let's say I want to compare two numbers that are stacked powers of different bases: $a^{b^{c^{d^e}}}$ compared to $f^{g^{h^{i^j}}}$ where all ten values will be integers in the range $[1,10]$. ...
39
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3answers
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A new interesting pattern to $i\uparrow\uparrow n$ that looks cool (and $z\uparrow\uparrow x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to ...
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2answers
112 views

Calculate $2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 2016$

How to find $2^{2^{2^{\cdot^{\cdot^{2}}}}} \mod 2016$ where $2$ occurs $2016$ times? My current observations: $$2^{11} = 2048 \equiv 2048=2016 \equiv 2^5 $$ and $$ 2^{16} \equiv 2^{11}\cdot 2^5 \...
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1answer
20 views

Differentiation equations with a power tower

I'm given that $$ x^{{mx}^{mx}...} = y^{{my}^{my}...}$$ I should find $ \frac {dy}{dx} $. How do I start? Is there any way to simplify this? For example, do the extra exponents stop mattering after ...
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1answer
117 views

$2^{3^{4^{…^{n}}}} \equiv 1$ (mod $n+1$)

I remember when I started learning modular arithmetics I found a tetration equation stated as follows $2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$) I am wondering how could this be proved, I tried this ...
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29 views

A different type of infinite power tower function

The question is about the following: Let a function be defined such that $$f_n(x) = x \uparrow \uparrow n$$ Where $n$ is a natural number Now, it is reported at many places that the function $$F_1(x)...
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1answer
43 views

Pattern in power towers of 2 involving last digits

We have \begin{align} 2^{2^{2}} &\mod 10 = 6 \\ 2^{2^{2^2}} &\mod 100 = 36 \\ 2^{2^{2^{2^2}}} &\mod 1000 = 736 \\ 2^{2^{2^{2^{2^{2}}}}} &\mod 10000 = 8736 \\ 2^{2^{2^{2^{2^{2^2}}}}} &...
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Supremum of arc lengths of graphs of power towers

Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only: $$x,\;x^x,\,x^{x^x},\;\...
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1answer
1k views

How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?

Suppose we have a power tower consisting of $2$ occurring $n$ times: $$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$ How many values can we generate by placing any number of parenthesis? It is fairly simple for ...
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1answer
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Attempt on fractional tetration

To start off, I was looking at the following ingeniously made form of the Gamma function: $$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$ which lies on the back of $$1=\...
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3answers
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Nested powers of $\sqrt 2$ has a solution different from its limit. What does this mean?

The infinitely nested power expression below has a limit of $2$: $$x=\sqrt2^{\sqrt2^{\sqrt2^{...}}}$$ In finding this limit, we may use: $$x=(\sqrt2)^x$$ But this expression has two solutions, $2$ ...
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1answer
55 views

Nested powers remainder problem.

I'm struggling a little with a question concerning power towers. I have this number $2018^{\large {2017}^{\Large 16050464}}\!$ and I want to find the remainder when it is divided by 1001. I have ...
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1answer
150 views

Smooth Elementary Function that Outgrows All Tower Functions?

This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer. Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth. Is ...
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1answer
87 views

Why is $x=4$ as a fixed point of a map $\sqrt{2}^{x}$ unstable?

My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x \mapsto f(x)$ is given, with $$f(x)= (\sqrt{2})^...
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2answers
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On the power tower $\exp(x-\exp(x-\cdots))$

The intention is to find the maximum of the power tower $\exp(x-\exp(x-\cdots))$. From here, we see that it is around $0.965$ or possibly even higher. The approximate value of its integral is also ...
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0answers
140 views

Partial Values for Knuth's Up-Arrows

$3 \uparrow 4 $ is $3^4$, and $3\uparrow \uparrow 3$ is $3^{3^{3^3}}$, etc. For those of you unfamiliar, here is a wiki page on the notation. Clearly, up-arrow expressions, as they are usually ...
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Is $i = e^{\frac{\pi}{2}e^{\frac{\pi}{2}^{.^{.^.}}}}$? [closed]

I came up with this and I am wondering if it is true, because it seems illogical that $i$ can be made from an infinite power tower of reals. The way I found this is the following: $$i=e^{\frac{\pi}{2}...
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3answers
328 views

Why is $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}} \approx\pi$ (matches upto 4 digits).

Is there something deeper here (like this is the truncation of an exact formula or something else going on)?
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3answers
256 views

What's a general algorithm/technique to find the last digit of a nested exponential?

So I'm working on this particular question at codewars, and asking this here because I've been trying to work it out for a day and a half now. The purpose: To find the last digit of a nested exponent ...
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4answers
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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 = x^{\Big(x^{x^{x^{...
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1answer
154 views

Area under the infinite tetration curve

What is the area under the curve where the infinite power tower converges? $$\lim_{y \to \infty} = {}^y x.$$ The formula for this curve is given by various sources as: $$\frac{\mathrm{W}(-\ln x)}{-\...
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1answer
158 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 names, ...
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1answer
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Solving for $x$ in $M ^ {M ^ M} = x ^ {1 / (x-1)}$ where $M = 5 ^ {\sqrt{5} / 10}$

Methods used by analogy, for example $x ^ x = 3 ^ 3 \implies x = 3$, Determine the value of $x$ in $$M ^ {M ^ M} = x ^ {1 / (x-1)}$$ if $M = 5 ^ {\sqrt{5} / 10}$.
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1answer
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Does the infinite power tower converge for all $1>x>0$

I know that if $$y=x^{x^{x^{x^{x\dots}}}}$$ then $$x=y^\frac1y$$ for values of $x$ where the infinite power tower converges, so when $x\le e^\frac1e$. However, when I put the power tower into ...
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1answer
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The sequence satisfying $a_2^{a_3^{\dots^{a_n}}} = n $

$a_2 = 2$ $a_2^{a_3 } = 3$ So $a_3= \ln(3)/\ln(2)$. I wonder about all solutions $a_n$ such that $a_2^{a_3^{\dots^{a_n}}} = n$ For all $n$. How does $a_n$ behave? What are the best asymptotics? ...
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Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
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Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Knuth arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)\\f(x,n)=x\...
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3↑↑↑3= ? but with 10 instead of 3 ( approximation, order of magnitude )

3↑↑↑3= (or near) in power tower of 10 or in ( Knuth ) arrow ↑ notation of 10 to get a sense of it's order of magnitude; I grasp numbers more easily with 10 3↑↑↑3 being the first really huge number in ...
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1answer
176 views

Solve $i^{i^{i^\ldots}}$ [duplicate]

How to find $$i^{i^{i^\ldots}} \quad :\quad i=\sqrt{-1}$$ I'm able to find the solution for the finite powers using $$i=e^{i(2k\pi+\frac{\pi}{2})}\quad:\quad k\in\mathbb{Z}$$ $$i^{i}=e^{-(2k\pi+\...
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1answer
171 views

What is the value of $i^{i^{i^\ldots}}$? [closed]

What is the value of $i^{i^{i^\ldots}}$? My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg ...
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2answers
406 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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2answers
561 views

Dividing power towers by exponents

Say we have $e^{e^{e^{e^e}}}$. Since exponents raised to exponents is the same as multiplying them, this is equivalent to $e^{4e}$: $$e^{e^{e^{e^e}}}=e^{4e}$$ Factoring out an $e^e$: $$e^{e^{e^e}}=...
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What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

What is the derivative of $$x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$$ My effort: Let $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}\implies g(x)=x!^{g(x)}$$ Taking natrual log on both ...
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1answer
55 views

The Tower of 5 5s

What are the last three digits of $5^{5^{5^{5^5}}}$? I tried using modular arithmetic, but it had fallen short. A detailed solution is greatly appreciated. Thank you very much in advance!
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$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ asymptotic?

Consider for positive real $x$ : $$ f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}} $$ How does this function behave ? How fast does it grow ? Faster than any fixed iteration of exp sure, but ...
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2answers
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Magnitude of $f_3(n)$ compared to power towers of tens

In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$ The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$ For example, to calculate $f_3(5)$, we have to ...
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3answers
171 views

Passing that ball on, where does it end up in? [duplicate]

$10$ people are seating on chairs around a circular table. These chairs are marked in a clockwise manner. There is a ball on the man’s hand who is seated on $0$ marked chair, and the ball will be ...
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1answer
100 views

Taylor series of a power tower

I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by $$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$ where $B_n$ are the Bell Numbers. However, I can't figure out a Taylor ...
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2answers
203 views

Shortcut to $x\uparrow \uparrow n$ for very large $n$ and $x\approx e^{(e^{-1})}$?

If the number $x$ is very close to $e^{(e^{-1})}$ , but a bit larger, for example $x=e^{(e^{-1})}+10^{-20}$, then tetrating $x$ many times can still be small. With $x=e^{(e^{-1})}+10^{-20}$ , even $x\...
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3answers
215 views

Power Towers, and Notation for Iterated Exponentiation

So far, we use the symbol $$\sum$$ to denote sums, and $$\prod$$ to denote products. But is there any such notation for exponentiation? Has any research been done about exponentiation of this type, ...
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1answer
62 views

Identify the base and exponent in $x^{x^x}$ in order to apply power rule of differentiation

While differentiating ${x}^{{x}^{x}}$ using power rule, what should be the base and exponent, i.e. base=$x$, exponent=$x^x$ or base=$x^x$, exponent=$x$. Any WHY?
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0answers
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Is $a^x\equiv x\mod 10^n$ always uniquely soveable, when $\gcd(a,10)=1$?

If $\gcd(a,10)=1$ and $n\ge 1$, is the equation $$a^x\equiv x\mod 10^n$$ always uniquely solveable modulo $10^n$ ? If yes, how can this be proven ? The discrete logarithm does not seem to help. I ...
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0answers
52 views

Tetration with 0 < #s < 1?

When I try numbers between 0 and 1 on my calculator app n-calc on my phone .. I get rapid convergence when the numbers are close to 1 .. And alternating but slow convergence when numbers are close to ...
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2answers
2k views

(Square root of 2) power (square root of 2) power…

The problem is to calculate A: A= sqrt(2)^sqrt(2)^sqrt(2)^... (Each one(not first and second!) is a power for the previous power) I used my usual(and only!) method: A=sqrt(2)^A It can't be correct ...
9
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3answers
362 views

Solutions of $a^{a^x}=x$ for fixed $a>0$

I am interested in the equation $a^{a^x}=x$ for some fixed $a>0$. Is there some way to rearrange for $x$ or solve otherwise? What about the nature of the solutions? For which fixed $a>0$ are ...