Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.

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.

73 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
14
votes
0answers
730 views

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 ...
11
votes
1answer
165 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 ...
6
votes
0answers
346 views

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{^} (...
5
votes
0answers
125 views

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. ...
5
votes
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
votes
0answers
246 views

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 =...
5
votes
0answers
331 views

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 $1$ ...
5
votes
0answers
200 views

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 ...
5
votes
0answers
790 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 ...
5
votes
0answers
271 views

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 ...
4
votes
2answers
82 views

tetration as an iterative solution for the transcendental equation $\sqrt[x]{x}=y$

for positive integer $n$ use the notation $y^{[n]}$ to represent the $n$-th tetration of $y$, so $y^{[1]}=y$, $\, y^{[2]}= y^y$, $\,y^{[3]}=y^{y^y}$, and so on. a few simulations suggest that on $(0,...
4
votes
0answers
92 views

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&...
4
votes
0answers
291 views

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 ...
4
votes
0answers
413 views

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 (...
4
votes
0answers
156 views

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 ...
3
votes
0answers
72 views

Tetration of cos : An original Inequality .

It's a little bit crazy for me the problem I propose to you today : Let $a,b,c\geq 0$ such that $a+b+c=1$ then we have : $$3\cos(\frac{1}{3})^{\cos(\frac{1}{3})^{\cos(\frac{1}{3})}}\geq\cos(a)^{\...
3
votes
0answers
147 views

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^\...
3
votes
1answer
169 views

Show that a certain parametric equation does not represent a cardioid

For $z \in \mathbb{C}$ consider the sequence $z, z^z, z^{z^z} ... ,$ that is, the iterated exponential with base $z$. If $z$ belongs to the Shell-Thron region the iterated exponential will converge. ...
3
votes
0answers
133 views

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{...
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. ...
3
votes
0answers
178 views

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}{...
3
votes
0answers
263 views

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 ...
3
votes
0answers
215 views

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. $a + a$, ...
3
votes
1answer
202 views

Tetration Binomial Theorem

I was exploring tetration and came across the following identities: $${^0}(ab) = 1$$ $${^2}(ab) = ({^2}a)^b * ({^2}b)^a$$ $${^3}(ab) = (ab)^({^2}(ab)) = (ab)^{(({^2}a)^b * ({^2}b)^a)}$$ That third ...
2
votes
0answers
19 views

Comparison of the Sizes Arbitrarily Generated Compound Exponential Numbers

I've been wondering about how it might be possible, given two compound exponential numbers generated by recurrence schemes equipped with addition multiplication and exponentiation (to some integer ...
2
votes
1answer
91 views

Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
2
votes
0answers
43 views

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}...
2
votes
0answers
122 views

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 ...
2
votes
0answers
76 views

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?
2
votes
0answers
64 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 ...
2
votes
0answers
230 views

$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) =...
2
votes
0answers
32 views

“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} {\...
2
votes
0answers
156 views

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 ...
2
votes
0answers
37 views

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 $^{0.5}2$, $^{3}3$, $...
2
votes
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
votes
0answers
145 views

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 ...
2
votes
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
votes
0answers
64 views

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 ...
2
votes
0answers
138 views

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)$...
2
votes
0answers
63 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 ...
2
votes
0answers
64 views

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 : ...
2
votes
0answers
33 views

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 $X$...
2
votes
0answers
99 views

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 ?
2
votes
0answers
115 views

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 ...
1
vote
0answers
40 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)...
1
vote
1answer
85 views

Prove that ${}^ni$ is complex for all $n \ge 3$

We can define ${}^nx$ as $\underbrace{\displaystyle {x^{x^{\cdot ^{\cdot ^{x}}}}}}_{n\text{ times}}$ (Tetration). I conjecture that ${}^ni$ is complex for all $n \ge 3, n \in \mathbb{N}$. I've ...
1
vote
0answers
122 views

Find all ordered triples of primes $(a, b, c)$ such that $a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$.

Find all ordered triples of primes $(a, b, c)$ such that $$\large a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$$ Notation: $$\large {^xy} \text{ or } x^{\underline y}$$ are nonations for ...
1
vote
0answers
63 views

Proving the derivation of the infinite tetration via Summation/Product notation is equivalent to the derivation via Implicit Differentiation

First some background info. Feel free to skip to the bold text below if you already know this. I) The derivative of the infinite tetration of $x$ (I guess that's how you say it) or ${}^\infty x$ ...
1
vote
0answers
32 views

Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
1
vote
0answers
28 views

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 &...