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.
401
questions
5
votes
1answer
175 views
What is the value of $i^{i^{i^\ldots}}$? [closed]
What is the value of $i^{i^{i^\ldots}}$?
My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg ...
2
votes
2answers
746 views
How to solve this tetration equation $\;^n 2 = \;^2 n $?
How would one find all real solutions to the following equation:
$\qquad$ $n^n = 2^{2^{2^{2^{\dots^2}}}} $(where the number of $2$s is equal to $n$)
generalizing to $n$ being a real value. In ...
1
vote
0answers
124 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 ...
5
votes
2answers
219 views
How fast do iterated exponentiation converge?
Iterated exponentiation is defined by
$$x \mapsto x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$$
or more conveniently, we denote by $^rx$ the expression $\underbrace{x^{x^{\cdot^{\cdot^{\cdot^{x}}}}}}_{r \text{ ...
6
votes
1answer
100 views
Can we use trigonometric idendities to calculate $\cos(x)$ and $\sin(x)$ for extremely large $x$?
If we want to calculate $\sin(x)$ and $\cos(x)$ for very large $x$ , lets say $10^5$ , the usual way is to reduce the number $x$ modulo $2\pi$.
If the number is a large power of a small number, for ...
6
votes
2answers
1k views
What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?
In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but ...
1
vote
1answer
222 views
The one value of super square root function
A function can not have more than one value.
i.e. we only take the positive value for $y$ where $y=\sqrt 4$
But what about the super square root?, if both value are positive real values like:
$$y=\...
3
votes
1answer
270 views
Prove if $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ than $z^{z^{z^{…}}}$ does not converge
Let $z \in \mathbb{C}$ and let $W$ be the Lambert W function. I am trying to prove that:
If $t = W(-\log z)$, $|t| = 1$, and $t^n \ne 1$ for all $n \in \mathbb{N}$ (ie, $t$ is not a root of unity) ...
1
vote
1answer
246 views
How would tetration work for non integer numbers.
Can you even do these and how would you do them? How does tetrations algebraically work?
$$^{.5}x=?$$
$$^{-1}x=?$$
$$^ix=?$$
Is there such a number like e that converges?
$$^xd=(some/equation/with/x)$$...
4
votes
1answer
413 views
Infinite tetration of $-2.5$
Let $a_n$ be the sequence $z, z^z, z^{z^z} ...$ for $z \in \mathbb{C}$. This is sometimes called the iterated exponential with base $z$. I am investigating the above sequence for $z = -2.5$. After ...
0
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1answer
258 views
Inverse operation of tetration and how it is computed?
If $c=a+b$, then $a=c-b$ and $b=c-a$. If $c=a\times b$, then $a=\frac{c}{b}$ and $b=\frac{c}{a}$. If $c=a^b$, then $a = \sqrt [b]{c} =c^{\frac{1}{b}}$ and $b=log_ac$.
What are the analogous inverse ...
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3answers
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What is this operator called?
If $x \cdot 2 = x + x$
and $x \cdot 3 = x + x + x$
and $x^2 = x \cdot x$
and $x^3 = x \cdot x \cdot x$
Is there an operator $\oplus$ such that:
$x \oplus 2 = x^x$
and $x \oplus 3 = {x^{x^x}}$?
...
-1
votes
1answer
112 views
Solve $(x+a)^{1/x} = b$ for $x$
Solve $(x+a)^{1/x} = b$ , for $x$
where $a$ & $b$ are real constant.
Do not use Lambert W-function in solution.
Instead of using Lambert W-function, there are solution steps look like "...
0
votes
2answers
115 views
$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$
From Numberphile $$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}}}} = 10^...
-1
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1answer
63 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
votes
1answer
228 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
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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
votes
1answer
158 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 ...
1
vote
1answer
150 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 on ...
3
votes
0answers
146 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:
$${\ ^{n}x = \ \atop {\ }} {{\underbrace{...
4
votes
3answers
205 views
The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$
A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$.
Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$
...
5
votes
1answer
118 views
Equality of power towers : $a\uparrow\uparrow m=b\uparrow \uparrow n$
Suppose, $a,m,b,n$ are natural numbers greater than $1$.
If we have $$a\uparrow\uparrow m=b\uparrow\uparrow n$$
can we conclude $a=b$ and $m=n$ ?
$a\uparrow \uparrow m$ is a powertower of $m$ $a's$ ...
3
votes
2answers
452 views
Tetration and Fractions
Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
0
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1answer
400 views
What is a puiseux series and what is wolfram-alpha doing with this antiderivative?
I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the ...
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3answers
1k views
Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”
$\DeclareMathOperator{\Arg}{Arg}$
Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
2
votes
1answer
194 views
Last three digits of tetration
Find the last three digits of the number:
$7^{7^{7^7...}}$
where there are 1001 sevens.
I know how to do it for when there are 4 and 5 sevens. I get an answer of 343. But how do I find it for ...
0
votes
0answers
175 views
Sum of the reciprocal of tetration?
Let $$f(x)=\sum^\infty_{n=1}\frac{1}{{}^xn}$$ where ${}^xn$ is n tetrated to the xth.
What are f(2) and f(3), and could you please also explain how you reached these answers?
3
votes
1answer
90 views
Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence
Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms.
But we can certainly compute roots, for example by this method.
...
0
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0answers
127 views
Showing that a Hermitian matrix can have eigenvalues that correspond to arbitrary numbers does not prove the Hilbert-Polya conjecture, does it?
I read in Wikipedia about the Hilbert-Polya conjecture that:
" ...a physical reason that the Riemann hypothesis should be true, and
suggested that this would be the case if the imaginary parts $...
3
votes
2answers
229 views
How fast does this sequence grow?
I have the following recursive definition of a sequence of numbers:
$$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 \...
3
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0answers
233 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:
$$ \frac{d\ ^nt}{dt} =\ ^nt \cdot log(t)\frac{d\ ^{n-1}t}{...
4
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3answers
777 views
Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent
Let $z \in \mathbb{C}$ and let $W$ be the Lambert $W$ function. In this post it is shown that if $|W(-\ln z)| > 1$ then the infinite power tower $z^{z^{z^{z^...}}}$ does not converge, that is $|W(-\...
6
votes
1answer
7k views
Why is exponentiation right associative? [duplicate]
From Wikipedia:
In order to reflect normal usage, addition, subtraction,
multiplication, and division operators are usually left-associative
while an exponentiation operator (if present) is ...
4
votes
1answer
769 views
How would you define non-integer tetration? [duplicate]
Tetration is defined for all $n\in \Bbb{N}$ by
$$
{^1}a = a \\
{^{n+1}}a = a^{\left({^n}a\right)}
$$
Thus ${^3}a$ means $a^{a^a}$.
Here $a$ could be any real (or indeed even complex) value, but only ...
2
votes
2answers
275 views
Do we know the value of $3 \uparrow\uparrow\uparrow 3$
I was studying Graham's number and before we can even start calculating $g_1$ which is:
$g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$,
I was wondering if we even have the actual value of:
$3 \...
2
votes
2answers
621 views
Convergence or divergence of infinite power towers of complex numbers $z^{z^{z^{z{…}}}}$
Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence $z$...
2
votes
2answers
817 views
How do I write Grahams number
I found that graham's number is :enter image description here
So, can we say that it is equal to $3^x$ with $x$ is a power tower of 63 3's?
3
votes
1answer
113 views
Question concerning comparison of different tetration functions
Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$
Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$
Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$?
If so, is the proof elementary? (Use only ...
2
votes
2answers
462 views
Integrate $x$ to the power $x$… to the power $x$… infinitely
This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it.
$$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$
How would you go about this?
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0answers
107 views
What the minimum of infinite tetration divided by $\sqrt{x}$?
For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point
$$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = \sqrt{2}^{\sqrt{2}^{\...
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4answers
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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 ...
2
votes
3answers
284 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
votes
1answer
805 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
votes
0answers
71 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.
$$y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)\\f(x,n)=x\...
0
votes
3answers
102 views
Finding the function that would describe this:
I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are:
1: 6/(x+1)
2: 8/(2^x)
3: 10/(?)
The question is, which one is next? I will say that ...
1
vote
2answers
284 views
Could you solve $xā^n2=xā^m2$?
As my title asks, could you solve $x^2=2^x$?
But that's the worrisome part, as I noticed $xā^n 2=xā^m 2$ and $2ā^p x=2ā^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
3
votes
4answers
405 views
Mathematical fallacy of $x^{x^{x^{x^x…}}}$ = 2
Suppose we have an equation with an infinite number of $x$'s as an exponent:
$$x^{x^{x^{x^x...}}} = 2$$
$$x^{(x^{x^{x^x...}})} = 2$$
because there are infinity $x$'s in the parentheses, which we've ...
2
votes
1answer
670 views
Exponential Factorial vs Tetration
I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
2
votes
1answer
155 views
Very confused about a limit.
This question is about where I made my mistake in the computation of a limit.
It relates to An answer I gave that confused me.
The question to which I gave the (partial) answer is related to ...
1
vote
0answers
32 views
Is $\max \int_{- \infty}^{\infty} \frac{dx}{\pi (1 + x^2 + f ' (x)^2 )} $ a uniqueness condition here?
Let f(x) be a real-differentiable function with $fā²(x)>0,fā²ā²(x)>0 $ and
$$ f(f(x)) = \exp(x) $$
for all real $x$.
Tommy1729 adds the optimization condition
$$ max \int_{- \infty}^{\infty} \...
