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Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

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Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
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How to solve $x^x = y^y \mod p$?

Let $p>5$ be a prime. How to solve $x^x = y^y \mod p$? How many solutions are there for a given $p$ such that $x,y < p$? I know the discrete logarithm and the theory of quadratic residues, but ...
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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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Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} (...
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function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$

Here it's cited: the existence of the holomorphic function ${\displaystyle \varphi }$ such that ${\displaystyle \varphi (\varphi (u))=\exp(u)}$ had been demonstrated in 1950 by Hellmuth Kneser. ...
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A limit related to asymptotic growth of tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
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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 ...
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Questions concerning the Integration of Integer Tetration

I've been interested in finding the antiderivative of integer tetration, a function defined as iterative exponentiation. Integer tetration is written as $^n$$x$ where $^1$$x =x$, $^2$$x =x^x$, $^3$$x =...
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Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
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Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$ The two smallest are $6771419$ and $72153167$, which I found by trial division. The number ...
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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 ...
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Arrow notation and decimals.

We know how to add with decimals. We know how to multiply with decimals. We know how to exponentiate with decimals. Do we know how to work with decimals for power towers? for example, can we deal ...
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New/useful method for summation of divergent series?

Questions $$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$ Also obeys (see background for argument): $$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
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Series of the form $\sum_{n=1}^\infty \frac{1}{ ^{s}n}$ where $^{s}n$ is the tetration of $n$

I am interested in the function $$T(s)=\sum_{n=1}^\infty \frac{1}{ ^{s}n}$$ where $s$ is an integer and $ ^s n$ is $n^{n^{n^{...}}}$ $s$ times. I know these series are convergent since tetration of $n&...
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When does $x^{x^{x^{…^x}}}$ diverge but $x^{x^{x^{…^c}}}$ converge?

Let us define these two sequences as follows: $a_0=1$, $b_0=c$ $a_{n+1}=x^{a_n}$, $b_{n+1}=x^{b_n}$ $c\ne x^c$ $x,c\in\mathbb C$ Is it possible for $a_n$ to diverge but $b_n$ to converge under ...
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Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation (...
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Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
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Tetration of a number giving a complex number

Giving this power equation: $$S=\lim_{n\to\infty} {^n}x=-i$$ where the symbol $^nx$ means the tetration operator, we can write in a form not formally correct: $${\ ^{n}x = \ \atop {\ }} {{\underbrace{...
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Derivation of tetration by iteration

I was screwing around a bit differentiating tetrations and was trying to write some rules for them. I came up with this recursive definition: $$ \frac{d\ ^nt}{dt} =\ ^nt \cdot log(t)\frac{d\ ^{n-1}t}{...
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Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
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Solve $x=d^{d}\log(d)$ using Lambert $W(x)$

We have $a=b^b$, so $$\log(a)=b\log(b)$$ $$x=\frac{x}{W(x)}\log\left(\frac{x}{W(x)}\right)$$ $$b=\frac{\log(a)}{W(\log(a))}$$ Next we have $c=d^{(d^{d})}$, so $$\log(c)=d^{d}\log(d)$$ In general $^{k}...
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Regularization of Exponentially exponential series?

Question What are the convergence properties of the last equation: $$K = e^{x} + x + \ln{x} + \ln\ln(x) + \dots $$ Can one artificially choose a value of $\ln (x)$ (since there is more than one ...
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How to solve $ax^x+bx+c=0$?

How can I solve $$ax^x+bx+c=0$$ or $$ax^{x^x}+bx^x+cx+d=0$$ where $x^x$ and $x^{x^x}$ - tetration? Is there analogue of discriminant for it?
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60 views

Is it hard to find out for which $r$ the infinite power tower $r\uparrow r \uparrow r\uparrow \cdots$ converges?

It is well known that the infinite power tower $$r\uparrow r\uparrow r\uparrow\cdots $$ with $r>0$ converges if and only if $e^{-e}\le r\le e^{1/e}$. I tried to prove it and I got stuck in the ...
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$f ^{\prime \prime}(x) = f(x) f^{\prime \prime} (x-1) $

I know the equation $$ f^{\prime} (x) = f(x) f^{\prime} (x-1) $$ is solved by $f(x) = C$ or by tetration ( $ f(x+1) = \exp(f(x)) $). So I wonder What are the solutions to $$g^{\prime \prime}(x) =...
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“Derivative-like” Operator such that $H(f(x)^{g(x)}) = H(f(x))^{H(g(x))}$

Given 2 functions $f(x),g(x)$ we have: $$D(f(x)+g(x)) = D(f(x))+D(g(x))$$ And $$D^*(f(x)g(x)) = D^*(f(x))D^*(g(x))$$ where $D^*$ is the multiplicative derivative: $$D^*(f(x)) = lim_{h\rightarrow 0} {\...
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How to find a function $f(x)$ such that $f(f(x))=\log_ax$?

Is there some method to do this or maybe some method to find a function such that $f(f(x))$ is at least approximately equal to $\log_{a}x$? Maybe the taylor series could be of help. So, we're looking ...
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What types of fractional tetrations are possible to calculate without using extensions of tetration?

For example, any number tetrated to a positive integer can be calculated by just doing repeated exponentiation from top to bottom. Similarly, it is possible to calculate values of $^{0.5}2$, $^{3}3$, $...
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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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Last three digits of tetration

Find the last three digits of the number: $7^{7^{7^7...}}$ where there are 1001 sevens. I know how to do it for when there are 4 and 5 sevens. I get an answer of 343. But how do I find it for ...
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68 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\...
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another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
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How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... $$(x!)\uparrow\uparrow(!x)$...
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59 views

Which primes satisfy this modular property?

Let $x$ be a residue$\mod p$ where $p$ is an odd prime. Im searching for such $p$ such that there exists a function $f(x)$ with propery $f(f(x)) - 2^x \equiv 0 \mod p $ for all values of $x$. I ...
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Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than $...
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Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
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Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton $X$...
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About growth rate of the iterated exponential on the complex plane.

Let $n$ be a positive integer. Let $f(n,x) = exp(f(n-1,x))$ and $f(0,x)=x$. Let $Q(f(n,x)) =1$ if $Re(f(n,x))<2$. Let $S(n,x) = \Sigma_{1}^{n} Q(f(n,x))$. How to estimate $S(n,2+i)$ efficiently ?
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Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$? I know this function has 2 other fixed points apart from $0$, so I'm not sure. Also ...
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Maximum of exponential tower

First I introduce a notation similar to $\sum_{i=1}^n a_i$ for exponentiation. I.e. for any (potentially infinite) sequence $a_i$ we define $$ ES_{i=1}^n a_i = \left\{\begin{matrix} a_1 &...
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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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29 views

Number of steps to get an error less than $\epsilon$?

The iteration $$x_1=r$$ $$x_{n+1}=r^{x_n}$$ with $r=e^{e^{-1}}$ tends to $e$. What is the smallest index $n$ such that $|x_n-e|<\epsilon$ ? For small $\epsilon$, it seems that the smallest ...
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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 ...
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Is this a good reasoning on fractional tetration?

Searching stuff on Wikipedia somehow I got to tetration, and got really interested on how could an interpretation of fractional tetration be given. So I did the following analysis $$ \left(a \cdot \...
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Weird result regarding “infinitely explosive” differential equations

Firstly, take the family of differential equations $\dot x = > \frac{dx}{dt}=x^\alpha$, for any $\alpha \in \mathbb R$ The solution to these equations is $$(\text{for } \alpha=1):x(t)=x_0e^t$$ $$(\...
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Is this a correct way to extend the definition of super-logarithms?

I think I may have found something new because it's giving the correct results. I'm using the functional-roots of $logx$ to calculate super-logarithms. You can read this post of mine to get the idea : ...
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65 views

Generalizing an iterative logarithm integral.

First, some notation: $f^{\star0}(x)=x$ $f^{\star k}(x)=f\left(f^{\star k-1}(x)\right)$ So that for any integer $k$, $\log^{\star k}(x)=\underbrace{\log(\log(\dots(\log}_k(x))\dots))$. I then came ...
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91 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}^{\...
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31 views

Is $\max \int_{- \infty}^{\infty} \frac{dx}{\pi (1 + x^2 + f ' (x)^2 )} $ a uniqueness condition here?

Let f(x) be a real-differentiable function with $f′(x)>0,f′′(x)>0 $ and $$ f(f(x)) = \exp(x) $$ for all real $x$. Tommy1729 adds the optimization condition $$ max \int_{- \infty}^{\infty} \...