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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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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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28 views

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
64 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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3answers
108 views

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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0answers
93 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 ...
2
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1answer
164 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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2answers
524 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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3answers
166 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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1answer
51 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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3answers
173 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)?
3
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1answer
120 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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0answers
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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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1answer
69 views

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? ...
6
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1answer
78 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 ...
2
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3answers
137 views

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

$10$ 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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2answers
103 views

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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2answers
127 views

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 ...
6
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2answers
186 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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1answer
49 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
40 views

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
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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 ...
9
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3answers
351 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 ...
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1answer
62 views

Defining the p-Frobenius for $GF(p^{16})$ explicit

Basics First let me tell you some background. Consider $p\equiv 13 \bmod 32$, therefore, $p\equiv 5\bmod 8$. That means, 2 is a none-quadratic residue in $GF(p)$. Therefore, we are able to build up $\...
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2answers
62 views

How does this equation follow from this?

Let $p>2$ be an odd number and let n be a positive integer. Prove that $p$ divides $$1^{p^n}+2^{p^n}+...+(p-1)^{p^n}.$$ Here is the solution: Define $k = p^n$ and note that $k$ is odd. Then \...
7
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3answers
159 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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2answers
142 views

Solve X=sqrt(A)^sqrt(A)^sqrt(A)^…infinty? [duplicate]

If $X= \newcommand{\W}{\operatorname{W}}\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{.{^{.^{\dots}}}}}}}}}}} $ then what is the value of $X^2-e^{1/X}$ ?
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2answers
93 views

Difference between calculator and google calc for power [duplicate]

I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3. Please explain me the difference between two ...
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0answers
68 views

Modular power tower mapping, is it injective?

Given an infinite sequence $a_1, a_2, \dots$ where all $a_i > 1$ we study $a_1^{\,a_2^{\,\cdots}} \bmod m$. While this is an infinite power tower that grows without bound, I argue that it can be ...
36
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2answers
1k views

A new interesting pattern to $i↑↑n$ that looks cool (and $z↑↑x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-...
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0answers
277 views

Compute modulo of prime power tower

I have a prime number $p$, and I need to compute $p$ ↑↑ $k$ mod $m$. here p ↑↑ k can be written as p ^ (p ^ (p ^ (p ... k times))) for example - $p = 5, k = 3, ...
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2answers
244 views

Infinite Power Tower

I've been having fun with the problem of finding the values of $n$ for which the infinite power tower $$\sqrt{2}^{\sqrt{2}^{...^{\sqrt{2}^n}}}$$ Has a finite value. My final answer was that it ...
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2answers
262 views

How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$?

How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$? In other words, how to find $ \underbrace{x^{x^{...^{x}}}}_k \mod m$, where $x$ is prime?
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0answers
132 views

How can I solve this infinite exponent tower?

Using calculus (or algebra), how would I solve an infinite exponent tower such as this? $$c_0x^{c_1x^{c_2x^{c_3x^{.^{.^.}}}}}=a$$ Where $c_0=1$ and $c_{n+1}=\frac{c_n}{2}$ for $n=0,1,\ldots$ and $a&...
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1answer
183 views

Find the last ten digits of this exponential tower.

I came across this challenge. Find the last ten (least significant) decimal digits of $x$ = 2^(3^(4^(5^(6^(7^(8^9)))))). First some notation. Let $x_n = n^{(n+1)^{(n+2)^{.^{.^{.^9}}}}}$ denote ...
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2answers
124 views

Confused with exponent rules?

What power would I need to raise $4^{2^{2^l}}$ to get $4^{2^{2^n}}$ where $n>l$? Very simple question but it has stumped me. I guess $2^{2^{n-l}}$ but I do not think that is right I am not sure?
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0answers
92 views

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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0answers
82 views

Fatou Coordinate with two complex conjugate fixed points and extending Tetration to real values

Lets us define $f(z)=z^2+z+c$ with real valued c>0 and iterate the function f(z). Then the Abel function for $f(z)$ is $$\alpha(z)\;\; \text{where}\;\; \alpha(f(z))=\alpha(z)+1$$ $$ f^{[\circ z]} = \...
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2answers
191 views

How to find $3^{3^{3^{\dots}}}\pmod{100}$?

I can show that $3^{3^{3^n}}\equiv7\pmod{10}$ since $3^1\equiv3\pmod{10}$ $3^2\equiv9\pmod{10}$ $3^3\equiv7\pmod{10}$ $3^4\equiv1\pmod{10}$ Thus, it reduces to $3^{(3^{3^n}\mod4)}$. I can then ...
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2answers
183 views

Power tower question

$$x^{x^{x^{.^{.^{.}}}}} = 8$$ Then how to solve for x? I first tried like this $x^8=8$ but I don't get any way to solve.
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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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6answers
223 views

Why is $2^{2^{2^n}}$ not equal to $16^n$?

Why is $a^{b^{c^d}}$ not equal to ${(a^{b^c})}^d$ (for positive n)? For example, WolframAlpha seems to say that $2^{2^{2^n}}$ is not equal to $16^n$.
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0answers
149 views

Peter Walker's $C^{\infty}$ conjectured nowhere analytic slog

Consider the function $h(x)$ which is conjectured to be $c^\infty$ nowhere analytic, and can be used to generate what we call the base change slog, which is the inverse of the basechange sexp. This ...
5
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1answer
154 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 ...
6
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4answers
171 views

How to calculate $p={{{{{{6^6}^6}^6}^6}^6}^6}$ $\mod7$?

I've tried two approaches: Approach 1 Since $6 \equiv -1 \pmod7$ So, $p=(-1)^t$ and $t$ is even Therefore, $p=1$. Approach 2 Since $6 \equiv -1 \pmod7$ So, $6^6 \equiv 1 \pmod7$. Hence, ...
5
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2answers
161 views

How fast do iterated exponentiation converge?

Iterated exponentiation is defined by $$x \mapsto x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$$ or more conveniently, we denote by $^rx$ the expression $\underbrace{x^{x^{\cdot^{\cdot^{\cdot^{x}}}}}}_{r \text{ ...
5
votes
2answers
335 views

What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?

In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but ...
2
votes
1answer
61 views

Evaluate $\frac{x^2}{y}$ where $x=a^{a^{a}}$ and $y=a^{a^{2a}}$

Evaluate $\frac{x^2}{y}$ where $x=a^{a^{a}}$ and $y=a^{a^{2a}}$ 1.$1$ 2.$x^{a^{a}}$ 3.$x^{1-a^a}$ 4.$x^{2-a^a}$ My solution: $x^2=a^{a^{a}}*a^{a^{a}}=a^{2a^{a}}$ $\frac{x^2}{y}=\...
1
vote
1answer
89 views

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 ...
3
votes
2answers
378 views

If $x^{x^{x^{16}}}=16$ calculate the value of $x^{x^{x^{12}}}$

I have done the following but I'not satisfied. if $x^a=a$ then by substitution follows that $x^a=x^{x^a}=x^{x^{x^a}}$ etc. So $x^{x^{x^{16}}}=16$ is equivalent to $x^{16}=16$, and $x=2^{\frac{1}{4}}$...
6
votes
3answers
400 views

Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?