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.

29 questions with no upvoted or accepted answers
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11
votes
1answer
164 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
7
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0answers
290 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
6
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0answers
84 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 ...
5
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0answers
68 views

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},\;\...
5
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0answers
157 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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0answers
787 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
4
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0answers
140 views

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 ...
4
votes
0answers
140 views

On the sequence $(f_n)$ defined by $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$

Consider the numbers $x^x$,$x^{(x^x)}$,$x^{(x^{(x^x)})}$, etc. Let $n$ be the number of times $x$ appears in the power tower and $f_n$ the corresponding function, for example $f_4(x)=x^{(x^{(x^x)})}$....
3
votes
1answer
88 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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0answers
56 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{...
2
votes
1answer
89 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})^...
2
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0answers
86 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]} = \...
2
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0answers
69 views

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\...
2
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0answers
161 views

Special power towers

Let $x(n)$ be the power tower $2 \uparrow 3 \uparrow 4 \uparrow \cdots \uparrow n$. Let $y(n)$ be the power tower $n \uparrow (n-1) \uparrow \cdots \uparrow 3 \uparrow 2$ My questions : Is there a ...
1
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0answers
47 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} $...
1
vote
1answer
57 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 ...
1
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0answers
141 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 ...
1
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0answers
87 views

$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ 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 ...
1
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0answers
51 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 ...
1
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0answers
60 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 ...
1
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0answers
179 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&...
1
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0answers
99 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}^{\...
1
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0answers
23 views

Sufficient condition for an equivalence

What is a sufficient condition for the equivalence $$ a_1 \uparrow a_2 \uparrow ... \uparrow a_n \equiv a_1 \uparrow a_2 \uparrow ... \uparrow a_n \uparrow a_{n+1}\ mod(\ m)\ ? $$ In a closely ...
0
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0answers
14 views

Is there a way to simplify a n times elevated number with a different number as the last one?

let me explain what I mean since the title might be pretty confusing: $x^{y^{y^{.^{.^i}}}}$ There could be any number of y and I need to write this in a smaller way. Keep in mind that "i" must be ...
0
votes
1answer
21 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 ...
0
votes
2answers
237 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 ...
0
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0answers
495 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, ...
0
votes
1answer
67 views

if -a^(-b^-c) is a positive integer and a, b, and c are integers, then…

(a) a must be negative (b) b must be negative (c) c must be negative (d) b must be an even positive integer (e) none of the above
0
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0answers
114 views

Condition for existence of solution to a power-tower equation.

"For two positive number a, b which satisfies the condition $\ln a \ln b <0$. equation $a^{b^{a^{b^x}}}=x$ has only one root if and only if ${\frac{d}{dx}a^{b^x}}_{x=t}\geq-1$, where $a^{b^t}=t$" ...