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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1answer
46 views

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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1answer
56 views

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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26 views

How I can prove or disprove the below claim regarding telescopic sum in exponent of tower?

let $v_n$ be a sequence defined as : $v_n=u_{n}-u_{n+1}$ and $u_n$ be a decreasing sequence such that $0<u_n <1$, I'm interested to prove or disprove the following claim : Claim :for $n\in \...
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1answer
46 views

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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2answers
60 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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18 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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2answers
83 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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1answer
165 views

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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65 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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1answer
59 views

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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33 views

Infinite power tower convergency

Is $$2^{(1+\frac 12)^{(1+\frac 13)^{\dots}}}$$ Convergent? And what is the limit of this?
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1answer
62 views

$i^i\in\mathbb R$

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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38 views

Power towers, sequences, fixed points, and a mysterious constant

Today I was thinking about the function $f(t)=t^{t^{t^{t...}}}$ Where $t>1$. We can define this notion more formally with a sequence, $a_{n+1}=t^{a_n}$ given $a_0=t$. If the sequence converges, say ...
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1answer
50 views

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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2answers
302 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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An example of nested radical and power tower .$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$

I want to share with you some of my last work: $$e^{-1}=(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{e^{-1}+(e^1-1)(e^{-1})^{\sqrt{\cdots}}}}}}$$ It's easy to solve using logarithm but I would like ...
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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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Comparison of power towers with different bases

If I define $T(c, k, r) = c^{c^{\cdot^{\cdot^{\cdot^{c^r}}}}}$ with $(k-1)$ $c$'s in the tower. I want to understand the asymptotic behaviour of towers with different values of $c$. I am specifically ...
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Special power tower $x^{\sqrt{x+x^{\sqrt{x+x^{\sqrt{\cdots}}}}}}$ and generalized Lambert's function

I'm interested by the following "nested radical-power tower" we have : $$x^{\sqrt{x+x^{\sqrt{x+x^{\sqrt{\cdots}}}}}}=S$$ My try : We have taking logarithm on both side : $$\ln(S)=\sqrt{x+S}\ln(x)$$ ...
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2answers
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Find the last digits of $a_{2009}$, and of $b_{2009}$.

Define the sequences $a_1, a_2,...$ and $*b_1, b_2,...*$ by $a_1 = b_1 = 7$ and $$a_{n+1} = {a_n}^7, \\ b_{n+1} = 7^{b_n}$$ for $n\ge 1$. Find the last digits of $a_{2009}$, and of $b_{2009}$. What ...
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45 views

Inequality of the April month $a^{\sqrt{b+b^{\sqrt{2b}}}}+b^{\sqrt{a+a^{\sqrt{2a}}}}\leq 1$

the main idea was to create an inequality based on the well know inequality of Vasile Cirtoaje and add some nested radicals it gives : Let $a,b>0$ such that $a+b=1$ then we have : $$a^{...
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1answer
170 views

Alternating power sequence

I quite randomly stumbled upon the following phenomenon: Let $ f:\mathbb R^+\to\mathbb R^+, x\mapsto x^{-2^{3^{-4^{\cdot^{\cdot^{\cdot}}}}}} $, then in the interval of $[1,20)$ the plot of $f$ looks ...
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53 views

Complexity of repeatedly applying Euler's totient

When performing modular power towers e.g. $a_0^{a_1^{a_2^{.^{.^.}}}}\bmod n$, Euler's totient theorem and it's generalization reduces the problem to computing $$n,\varphi(n),\varphi(\varphi(n)),\...
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123 views

How to compute $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!\!\bmod 46,$ for power tower height $2020$?

What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$? The level of powers is $2020$. First there is no parenthesis so it means 3 power of 3 which is also power 3 ...
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3answers
114 views

How to solve $x^{x^{x^{x^{2010}}}} = 2010$

So I know that there is a difference between $(x^2)^3$ and $x^{2^3}$. But how do I use this knowledge to solve $$x^{x^{x^{x^{2010}}}} = 2010?$$
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1answer
58 views

Simple power tower $x^{1-x+x^{1-x+x^{1-x+x^{1-x+\cdots}}}}=x$

I was not clear in my last post : Let $0<x$ a real number then we have : $$x^{1-x+x^{1-x+x^{1-x+x^{1-x+\cdots}}}}=x$$ I take the logarithm of both side we get : $$\ln(x)({1-x+x^{1-x+x^{1-x+x^...
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0answers
69 views

Sum of power tower which tends to $1$

Inspired by an inequality of Vasile Cirtoaje, I have this : Let $0<x<1$ then : $$x^{2(1-x)}+(1-x)^{2x}\leq 1$$ Let $0<x<1$:$$f(x)=x^{1-x+x^{1-x+x^{2 (1-x)}}}$$ Then : ...
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46 views

Non integer Hyper-powers. [duplicate]

If I have a function $y=x^x$, that can be denoted in hyperpower notation as $^2x$, but I will be denoting it as $ y= $ hyp$_2(x) $. In general, for hyperpowers, $y=x^{x^{x^{...}x}}$ or in my notation $...
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1answer
63 views

Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is ...
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1answer
127 views

Proving a modified Ackermann function using induction

So I have 2 functions: $$ a(0,y) = y+ 1\\ a(x, 0) = a(x-1,1)\\ a(x,y) = a(x-1,a(x,y-1)) $$ And: $$ c(0,n) = 0\\ c(1,n) = n^2 + n +1\\ c(m,0) = c(m-1,1)\\ c(m,n) = c(m-1,c(m,n-1)) $$ And then I have:...
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2answers
173 views

Ackermann-Péter Function: Proof by induction, power tower

So I have this Ackermann function: $$ A(0,y) = y+ 1\\ A(x, 0) = A(x-1,1)\\ A(x,y) = A(x-1,A(x,y-1)) $$ And I have another function: $$ d(n) = 2^{2^{.^{.^{.^{.^{2}}}}}} $$ while the height of the ...
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50 views

How to find the maximum arc length of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ and the value of $r$ at which it occurs?

After seeing a discussion about graphs of the relationship $x^x + y^y = r^r$, it got me interested in attempting to see what the graphical appearance of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ would ...
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0answers
57 views

How to evaluate the infinite sequence of 0^(-1)^(-2)^(-3)^(-4)^{.}^{.}^{.}?

(Note: I deliberately didn't write the title in Latex because the notation doesn't display properly with power towers. Specifically regarding the ellipsis, which displays as "..." instead of ".^{.}^{.}...
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1answer
114 views

Determine the smallest positive integer $ m $ for which $\underbrace{100^{100^{\ldots^{100}}}}_m>\underbrace{3^{3^{\ldots^3}}}_{100}$

The functions $ f $ and $ g $ are defined by $ f (x) = 3 ^ x $ and $ g (x) = 100 ^ x $. Two sequences $ a_1, a_2, a_3, \ldots$ and $ b_1, b_2, b_3, \ldots $ are then defined as follows: (i) $ a_1 = 3 $...
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2answers
75 views

How to express the infinite power tower $a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$?

Just a relatively simple question; I'm just wondering what would be the proper notation to use to express an infinite power tower that has each repeated exponent increasing by a value of $1$, like ...
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1answer
287 views

Area enclosed by a function containing two power towers: $f(x)=(−\ln(x↑↑(2k)))↑↑(2k+1)$

I've been considering functions involving power towers lately and come across the following function: $$f(x)=(−\ln(x↑↑(2k)))↑↑(2k+1)$$ $$\text{Where }k∈\mathbb{Z} ^+$$ In the image below we can see ...
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1answer
50 views

What will be the minimum value of the function $f(x)=\arcsin(x)\uparrow\uparrow(2k)$ as $k \to \infty$?

Consider the following two functions involving power towers; $$f(x)=\arcsin(x)\uparrow\uparrow(2k)$$ $$g(x)=\arcsin(x)\uparrow\uparrow(2k-1)$$ Where $k\in \mathbb{Z}^+$. The global minimum of $f(x)$ ...
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3answers
128 views

Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$

I'm a computer student, learning math just for fun. Today I was graphing for fun that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to ...
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1answer
67 views

Nested Tetration properties

Regarding tetration, I know properties like ${}^a({}^bn)= {}^{ab}n$ do not hold in general. When $a=b=2$, for instance, we have $$ {}^2({}^2n)={}^2\left(n^n \right)=\left(n^n \right)^{\left(n^n \right)...
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1answer
60 views

The factors of a tetration plus an integer

There I was, just messing around with tetration, when I stumbled across this - $(x^x +1)/(x+1)$ = integer (for odd integer values of x) Playing some more with this it seems (not entirely sure as ...
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2answers
116 views

Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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1answer
71 views

How to find the last digit of a number in base b?

For a number $a^{x^{...^n}} $. To find its last digit in a base b, Imagine that I have this number $a^{x^{y}}$ to simplify the problem. Then I calculate $a^{x} \equiv c \pmod b$ and after that $c^{...
3
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1answer
139 views

Uniquely extended fractional iterations of $\exp$

Let us define the following basic conditions for an iterated exponential function: $$\exp^1(x)=e^x\tag{$\forall x$}$$ $$\exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$}$$ I then pondered what ...
3
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1answer
138 views

Compare power towers

Prove or disprove: $3^{3^{3^{3^{3...^3}}}}$ with 100 threes $>4^{4^{4^{4^{4...^4}}}}$ with 99 fours. Taking logs is useless, and there seems to be no other way to compare. Thanks!
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0answers
60 views

Solving congruence with powers

Consider that $c$ is integer that is $\geq \ 3$, also $d$ is non negative integer. How to figure out general formula for congruence: $$ \left( 2^{\large 2\cdot3^{c-3} +\large d}\right)\bmod 3^{c-1} $...
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2answers
101 views

The units digit of $(((\dots((2018^{2017})^{2016})^{.^{.^{.}}})^3)^2)^1$

I posted a problem, I got the answer from many guys, thanks for them. This is another problem, I am curious how to solve it. I tried to use modular arithmetic as in the problem linked above, but I ...
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6answers
155 views

The units digit of a power tower of consecutive numbers, from 2019 to 1

Is it possible to find the units digit of $2019^{2018^{2017^{.^{.^{.^{3^{2^{1}}}}}}}}$? Where the expression contains all natural numbers $[1,2018]$ as powers and $2019$ as the main base. Any help ...
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2answers
126 views

How iterated exponential $\exp^{[\circ x]}(y)$, $y\neq 1$, defined based on tetration?

Background: The tetration \begin{equation} ^xe = \exp^{[\circ x]}(1) = \underbrace{e^{e^{\cdot^{\cdot^e}}}}_{x \text{ times}} \end{equation} is well defined when $x \in \mathbb{Z}$. The extension of ...
3
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1answer
93 views

The digital root of a tower of exponents, $d(\underset{\text{The number of }2 \text{'s is }2013}{\underbrace{2^{2^{2^{.^{.^{.^{2}}}}}}}})$

For a natural number $n$, the digital root of $n$ is the value obtained by an iterative process of summing digits. The digital root of $n$ is denoted by $d(n)$. Examples; $d(142)=7$, $d(123785)=8$ ...
4
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1answer
127 views

Does $x^{x^{x^{x^{x^{\,\,\style{display: inline-block; transform: rotate(60deg)}{\vdots}}}}}} =y$ imply $2=4$? [duplicate]

Edit: My question has been requested to close due to its apparent lack of clarity. My question is below under "Problem". If the information above it is redundant, please let me know in a comment. I ...