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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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4
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1answer
393 views

Infinite tetration of $-2.5$

Let $a_n$ be the sequence $z, z^z, z^{z^z} ...$ for $z \in \mathbb{C}$. This is sometimes called the iterated exponential with base $z$. I am investigating the above sequence for $z = -2.5$. After ...
0
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1answer
234 views

Inverse operation of tetration and how it is computed?

If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$. What are the analogous inverse ...
28
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3answers
4k views

What is this operator called?

If $x \cdot 2 = x + x$ and $x \cdot 3 = x + x + x$ and $x^2 = x \cdot x$ and $x^3 = x \cdot x \cdot x$ Is there an operator $\oplus$ such that: $x \oplus 2 = x^x$ and $x \oplus 3 = {x^{x^x}}$? ...
-1
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1answer
106 views

Solve $(x+a)^{1/x} = b$ for $x$

Solve $(x+a)^{1/x} = b$ , for $x$ where $a$ & $b$ are real constant. Do not use Lambert W-function in solution. Instead of using Lambert W-function, there are solution steps look like "...
0
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2answers
114 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
-1
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1answer
59 views

separate $a$ & $b$ in $ssrt(a^a*b^b)$ [closed]

It is already known that $ssrt(a^a*b^b)$ does not equal $ssrt(a^a)*ssrt(b^b) = a*b$ Is there any other method to separate $a$ and $b$? ****Please note that $ssrt$ is "super square root". and my ...
0
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1answer
183 views

If $|t| = |W(-\ln z)| = 1$ and $t^n =1$ then $z^{z^{z^{…}}}$ is convergent

Let $z \in \mathbb{C}$ and $W$ be the Lambert W function. In this post I was told if $|t| = |W(-\ln z)| = 1$ and $t^n =1$ for some $n \in \mathbb{N}$ than the iterated exponential $z^{z^{z^{...}}}$ ...
12
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1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
4
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1answer
148 views

How can I show, that $N\uparrow\uparrow N$ is not “much larger” than $N$ for very large $N\ $?

Here : https://sites.google.com/site/largenumbers/home/3-2/knuth Saibian demonstrates that for very large numbers $N$, $N\uparrow\uparrow N$ is only "slightly larger" than $N$. I would like to ...
1
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1answer
137 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp\left(\frac{-K \cdot (m - a(n))}{m}\right),\ n \geq 1$?

Edit: In the original post, I put the function $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ which is not the function I wanted to study. The correct one is the one given below I came up ...
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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{...
4
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3answers
187 views

The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
5
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1answer
113 views

Equality of power towers : $a\uparrow\uparrow m=b\uparrow \uparrow n$

Suppose, $a,m,b,n$ are natural numbers greater than $1$. If we have $$a\uparrow\uparrow m=b\uparrow\uparrow n$$ can we conclude $a=b$ and $m=n$ ? $a\uparrow \uparrow m$ is a powertower of $m$ $a's$ ...
3
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2answers
330 views

Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
0
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1answer
294 views

What is a puiseux series and what is wolfram-alpha doing with this antiderivative?

I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the ...
10
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3answers
862 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
2
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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 ...
0
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0answers
143 views

Sum of the reciprocal of tetration?

Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth. What are f(2) and f(3), and could you please also explain how you reached these answers?
3
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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. ...
0
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0answers
117 views

Showing that a Hermitian matrix can have eigenvalues that correspond to arbitrary numbers does not prove the Hilbert-Polya conjecture, does it?

I read in Wikipedia about the Hilbert-Polya conjecture that: " ...a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts $...
3
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2answers
167 views

How fast does this sequence grow?

I have the following recursive definition of a sequence of numbers: $$a_{n+1}=(a_n)^{(a_{n-1})}$$ And $a_0=a_1=2$. The first few terms are: $$a_2=4$$ $$a_3=16$$ $$a_4=65536$$ $$a_5=1.1579209 \...
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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}{...
4
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3answers
684 views

Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent

Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^...}}}$ does not converge, that is $|W(-\...
5
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1answer
6k views

Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
3
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1answer
544 views

How would you define non-integer tetration? [duplicate]

Tetration is defined for all $n\in \Bbb{N}$ by $$ {^1}a = a \\ {^{n+1}}a = a^{\left({^n}a\right)} $$ Thus ${^3}a$ means $a^{a^a}$. Here $a$ could be any real (or indeed even complex) value, but only ...
2
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2answers
248 views

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham's number and before we can even start calculating $g_1$ which is: $g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of: $3 \...
2
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2answers
540 views

Convergence or divergence of infinite power towers of complex numbers $z^{z^{z^{z{…}}}}$

Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence $z$...
2
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2answers
612 views

How do I write Grahams number

I found that graham's number is :enter image description here So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?
3
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1answer
107 views

Question concerning comparison of different tetration functions

Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$ Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$ Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$? If so, is the proof elementary? (Use only ...
2
votes
2answers
411 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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0answers
100 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}^{\...
35
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4answers
1k views

An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?

Start with $i=\sqrt{-1}$. This will be $a_1$. $a_2$ will be $i^i$. $a_3$ will be $i^{i^{i}}$. $\vdots$ etc. In Knuth up-arrow notation: $$a_n=i\uparrow\uparrow n$$ And, amazingly, you can ...
3
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3answers
226 views

Infinite exponentials

We can read a lot of about convergence of series or Infinite products. E.g. for series. Following series $$\sum_{i=1}^\infty a_i$$ is convergent when $$\lim_{n\rightarrow\infty}a_n=0$$ and D'...
2
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1answer
685 views

Infinite tetration of $z$ where $z=i^i$

1) Proof that $z=i^i$ is a real number Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power ...
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\...
0
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3answers
100 views

Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
1
vote
2answers
254 views

Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
3
votes
4answers
353 views

Mathematical fallacy of $x^{x^{x^{x^x…}}}$ = 2

Suppose we have an equation with an infinite number of $x$'s as an exponent: $$x^{x^{x^{x^x...}}} = 2$$ $$x^{(x^{x^{x^x...}})} = 2$$ because there are infinity $x$'s in the parentheses, which we've ...
2
votes
1answer
495 views

Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
2
votes
1answer
147 views

Very confused about a limit.

This question is about where I made my mistake in the computation of a limit. It relates to An answer I gave that confused me. The question to which I gave the (partial) answer is related to ...
1
vote
0answers
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} \...
8
votes
4answers
379 views

A limit related to super-root (tetration inverse).

Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ (...
1
vote
1answer
144 views

How do I evaluate this summation?

I was wondering how do I solve the summation? $$\frac{1}{n}\sum_{j=1}^nja_{m,j-1}a_{m-1,j-1}$$ I found it in this link on power towers http://mathworld.wolfram.com/PowerTower.html
11
votes
2answers
443 views

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 ...
0
votes
1answer
896 views

Infinite tetration convergence

I came across infinite tetrations on wikipedia (https://en.wikipedia.org/wiki/Tetration) it says that the infinite tetration converges if and only if $\ e^{-e} \leq x \leq e^{1/e}$. I was wondering if ...
5
votes
2answers
103 views

Finding the Final Digits in a Stacked Exponents Problem

How to find the the last two digits of the number $$\underbrace{\huge 7^{7^{7^{...}}}}_{1+n}$$ When there are $n$ sevens in the ascending exponents. I have no idea how to condense this or where ...
-2
votes
1answer
75 views

Calculate this power? [closed]

Let $a$ is a positive real number. If ${a^{{a^{{a^{16}}}}}} = 16$ how much is ${a^{{a^{{a^{12}}}}}}$?
5
votes
2answers
95 views

Checking primality for $2 \uparrow \uparrow n + 3 \uparrow \uparrow n$

Is there a clever way to test primality for $$2 \uparrow \uparrow n \quad + \quad 3 \uparrow \uparrow n$$ where $n \gt 3$? (Not surprisingly, I got stuck after that) For $n \le 3$ we get: $$ \...
4
votes
5answers
348 views

Finding ${i^i}^{i\cdots}$ [duplicate]

We know that surprisingly enough, $i^i=\frac1{e^{\frac\pi2}}$. But what about finding the value of ${i^i}^{i\cdots}$? Is it possible? My attempt: Let $${i^i}^{i\cdots}=x$$ $$i^x=x$$ Or $$\left(\cos\...
1
vote
1answer
108 views

How can i count number of digits in tetrated numbers?

As you read in the title, I need a technique for counting number of digits in tetrated numbers. For example: ${3^{(3)}}^3 = 7625597484987 $(13 digits) ${7^{(7)}}^7 = $How many digits (...