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.

Filter by
Sorted by
Tagged with
4
votes
2answers
70 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 ...
1
vote
2answers
126 views

Convergence of the sequence $x_{n+1} = a^{x_n}$ [duplicate]

Let $a > 0$. Show that the sequence defined by $$ x_0 = 1, \qquad x_{n+1} = a^{x_n} $$ converges for $a \leq e^{1/e}$. Any help is appreciated, I don't even know where to start with this. Edit to ...
0
votes
1answer
42 views

Finding the derivative of $x$ tetrated to the $x$

Differentiating the functions $x^x$, $x^{x^x}$ (or ${^2{x}}$ and ${^3{x}}$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this ...
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
55 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 ...
3
votes
3answers
91 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 ...
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 ...
2
votes
1answer
41 views

inverse tetrations of complex numbers

for the complex function $f_{(2)}(z)=z^z$,where in the complex plane does the inverse $z_{(2)}(f)$ not exist, same for inverse of functions $f_{(3)}(z)=z^{z^z}$ being $z_{(3)}(f)$ and so on for ...
3
votes
2answers
90 views

Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges?

let $f_1(x)=f(x)$ $f_2(x)=f(f(x))$ $f_3(x)=f(f(f(x)))$ and so on... Is there some function $f(x)$ for which $f_{\infty}(x)$ is a continuous non-constant function that converges? It is okay if the ...
3
votes
1answer
125 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
votes
1answer
113 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!
0
votes
3answers
42 views

When raising to a power and tetrating, which comes first? Is $^24^3$ equal to $(^24)^3$, or to $^2(4^3)$?

I know tetration isn't quite an used operation, but anyway, what if it were featured in an expression? For example, what does $^24^3$ mean? Is it $(^24)^3=(4^4)^3=4^{12}$ or is it $^2(4^3)=^2{64}=64^{...
2
votes
2answers
108 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 ...
4
votes
2answers
208 views

Proof (or hints towards proof) for asymptotic shape of orbit $0 \to 1 \to b \to b^b \to \cdots$ with certain class of $b$?

In a paper by Baker & Rippon (1983) the property of being convergent or divergent for iterated exponentials $z_{h+1} \to b^{z_h}$ with $b$ complex and $z_0=1, z_1=b, z_2=b^b, \cdots$ for classes ...
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 ...
5
votes
3answers
175 views

Calculating the nth super-root when n is greater than 2?

Tetration (literally "4th operator iteration") is iterated exponents, much like how exponents are iterated multiplying. For example, $2$^^$3$ is the same as $2^{2^2}$, which is $4^2$ or 16. This ...
0
votes
1answer
50 views

Find range of $a$ if $\{x_n\}$ converges where $x_{n+1}=a^{x_{n}},n=1,2,\ldots$ and $x_1=a$ [duplicate]

For $a>1, x_{n+1}=a^{x_{n}}, n=1,2, \ldots$ and $x_{1}=a$. If $\{x_{n}\}$ converges, find range of $a$. $\begin{aligned} x_{n+1}-x_{n}&=f\left(x_{n}\right)-f\left(x_{n-1}\right)\\ &=f^{\...
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 ${}^\infty x$ ...
12
votes
1answer
286 views

What is the maximum of $\sum_{k=1}^{\infty} (-1)^k(^kx)$?

During my testing of the series $\sum\limits_{k=1}^{n} (-1)^k(^kx)$, I found that the sum converges to two limits when $n \to \infty$, for $e^{-e} \lt x \le e^{1/e}$ and oscillates between depending ...
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)^{\...
2
votes
1answer
135 views

What Tetration Algorithm can I utilize

I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow": A mandelbrot fractal (I call it "operation order 1") is done by ...
1
vote
2answers
74 views

Interpolation between iterations of exponentials (and logarithms)

I'm interested in finding a continuous and ideally smooth family of real-valued monotonic functions $H_t(x)$ that interpolate between iterations of exponentials, and also iterations of logarithms. So ...
0
votes
1answer
54 views

Is there a “natural tetration function”?

For the natural exponential function, $$ f(x)=e^x \to f(x)=f'(x). $$ Is there a natural tetration (tetral?) function? $$ f(x)={{^x}b} \to f(x)=f'(x) $$ Is it base $e$?
2
votes
3answers
209 views

How to compute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $ \int{x↑↑n} dx $ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
2
votes
1answer
101 views

Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
0
votes
1answer
61 views

Continuous, infinitely differentiable tetration map

We know that the exponential map is infinitely differentiable; is there a tetration map $TET$ defined on $(0,\infty)$ such that 1) $TET(x) > TET(y)$ when $x > y$ 2) $TET(e^x) = TET(x) +1$ 3) $...
-1
votes
1answer
25 views

Power tower(infinite tetration) [duplicate]

How to find the domain and range of $Y=x^{x^{x^{x^{x^{...}}}}}=x^Y$ I know how to differentiate the function. I don't know how to proceed further. We should prove that domain:$[1/(e^e),e^{(1/e)}]$ ...
1
vote
1answer
55 views

Pattern in power towers of 2 involving last digits

We have \begin{align} 2^{2^{2}} &\mod 10 = 6 \\ 2^{2^{2^2}} &\mod 100 = 36 \\ 2^{2^{2^{2^2}}} &\mod 1000 = 736 \\ 2^{2^{2^{2^{2^{2}}}}} &\mod 10000 = 8736 \\ 2^{2^{2^{2^{2^{2^2}}}}} &...
7
votes
2answers
102 views

Associocommutativity

One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring ...
3
votes
1answer
172 views

Smooth Elementary Function that Outgrows All Tower Functions?

This started off with me goofing off on Twitter and quickly led to this question to which I didn't know the answer. Let $T_2(x) = x^x$, $T_3(x) = x^{x{^x}}$, $T_4(x) = x^{x^{x^x}}$, and so forth. Is ...
0
votes
1answer
99 views

What is the name for this: “x^^y” (as in 2^^2=2^2 and 3^^3=3^3^3 and so on)

How do you call / is there any specific name for the following: x^^y e.g.: 2^^2 = 2^2 3^^3 = 3^3^3 4^^4 = 4^4^4^4 ...
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 ...
3
votes
1answer
109 views

Can we obtain an explicit and efficient analytic interpolation of tetration by this method?

I am curious about this. It has been a very long time since I have ever toyed with this topic but it was an old interest of mine quite some time ago - maybe 8 years (250 megaseconds) ago at least, and ...
5
votes
1answer
81 views

An Integral Designed to be on the Very Cusp of Convergence

Say we have an integer $n$∊ℕ₀ & a sequence of $n+1$ real numbers $\alpha_k\in[0,\infty)∀k$, where $k=0\dots n$, and using $\log^{[k]}$ to denote $k$ functionings of the logarithm ($\log^{[0]}x\...
2
votes
4answers
165 views

Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher order ...
2
votes
1answer
91 views

Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
3
votes
2answers
83 views

Tetration of non-integers: is there something wrong with this approach?

I'm trying to figure out a formula for tetration that will work for non-integer heights. I know the usual recurrence relation for tetration ($x \in \mathbb{R}, \text{ }n \in \mathbb{N})$: $${^{n}x} =...
0
votes
0answers
35 views

Is There an Operation Between Exponentiation and Tetration?

The generalisation of exponentiation is tetration, which is just repeated exponentiation. If we denote exponentiation as \begin{align*} \text{exp} (a, n) = a^n=a\cdot a\cdots a \end{align*} and ...
0
votes
0answers
50 views

The Infinite Tetration of x? [duplicate]

The infinite tetration of ${x}$ basically means that we take a value and we continue to raise the value to its power forever. If that sounds confusing, it can be thought of as infinite repeated ...
0
votes
0answers
26 views

quadratic tetraic equation

I have recently found interesting thing: Lambert W funtion inverse of $f(x)=xe^x$ it was easy to find roots of $x^x=a$ but I wonder is possible to find formula for root of equation $ax^x+bx+c=0$ ????
0
votes
2answers
37 views

Larger value with right associative tetration?

Given right associative tetration where: $^{m}n =$ n^(n^(n^…)) And a situation such as: $^{m}n = y$ $^{q}p = z$ What is a practical way to calculate which of $y$ and $z$ are larger? I'm ...
0
votes
2answers
124 views

Problem with derivative of $x^{x^x}$

I was recently watching blackpenredpen’s video (found here: https://m.youtube.com/watch?v=UJ3Ahpcvmf8) where he found the derivative of the the function $y = x^{x^x}$. Before watching the video, I ...
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}...
0
votes
1answer
90 views

Tetration induction proof

The triple arrow-up denotes power towers in which the number of levels themselves is a power tower with a number of levels that is a power tower, and so on. For example, $$\begin{align} a\uparrow\...
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 &...
2
votes
1answer
58 views

Concise notation for iterated exponentiation involving an unknown

I am working with some tetration problems, such as below: $$y = e^{e^x}$$ and I am looking for a concise notation for this. In particular, I would like a way to indicate $n$ iterations of the ...
10
votes
1answer
146 views

Function $f$ s.t. $\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$

The questions are: 1) Does there exists some function $f$ s.t. $\lim_{x\to\infty}\frac{f(e^x)}{f(x)}=1$ and $\lim_{x\to\infty}f(x)=\infty$? 2) Is $\big(\sum_{k=n}^{2^n}a_k\big)_n\to0$ is ...
2
votes
1answer
180 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+\...
0
votes
0answers
48 views

Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed?

Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper): $$ \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...