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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Growth of factorial functions and tetration
Good morning everyone, I have a doubt about the growth rate of the following two functions:
$$ \operatorname f(x)=x! $$
and
$$ \operatorname g(x)=^x n,\quad\mbox{where}\ \quad n \in \mathbb{R^+} $$
I ...
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Is $x^x = (x-1)^{x+1}$?
Background: I was trying to estimate the size of $21^{21}$ for some problem and decided to use $20^{22}$ as hopefully a rough approximate ($20^{22} = 2^{22} \cdot 10^{22} \approx 10^{28}$). But then I ...
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Solution of $x^x = (x-1)^{x+1}$ [closed]
Is there any way to solve this equation algebraically and give an exact form of the solution: $x^x = (x-1)^{x+1}$? WolframAlpha only finds the approximate solution 4.14.
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How to interlopate the pickover factorial?
Question
I want to be able to interlopate the pickover factorial, defined as: $$n$ = ^{n!}n!$$ where $^xx$ represents tetration. I want to be able to interlopate this function.
Context
The reason I ...
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Could an equation like ${^{h}b} = b^h + $c together with polynomial interpolation be used to calculate tetration? Are there flaws in it?
As it is quite easy to calculate integer solutions for $\;^hb = b^h $ the question arose on how to find a solution for real heights.
One idea was of predicting the height by finding a formula ...
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Finding the convergent interval of $z^{{2z}^{{3z}\ldots^{{nz}\ldots}}}$
I was trying to find an exact solution to $x=a^x$, and naturally derived a solution defined by infinite tetrations of $a$. I first defined a recursion equivalent to the definition of tetrations and ...
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Contradiction occurred trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ [closed]
Today I was trying to calculate the value of $\sqrt{2}+Re\Bigl(^{10}\hspace{-1mm}\left(-\sqrt{2}\right)\Bigr)$ (i.e., $\sqrt{2}$ plus the real part of the tenth tetration of the base $-\sqrt{2}$) up ...
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Infinite exponentiation [duplicate]
It started as an exercise for my students.
Calculate $i^i$, then $i^{i^{i}}$ and make a conjecture if we follow that pattern.
If we define $u_n=i$ and $$u_{n+1}=i^{z_n}=e^{i\frac{\pi}{2}z_n}$$
Then, ...
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Tetration uniqueness by $ A = \inf \sum_n a_n^2 $?
Let $x > -2$ and $f(0) = 1,f(x+1) = \exp(f(x))$
And $f$ is a taylor series :
$$f(x) = \sum_n a_n x^n$$
where the $a_n$ are all real and all nonzero and $f$ has radius $2$.
(Notice $f(-1) = 0, f(-2) ...
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Exponential Fibonacci and its recurrence-relation to ϕ. [closed]
1,1,2,3,5,8,... = Additive Fibonacci: a(n-1) * phi is asymptotic to a(n)
2,2,4,8,32,256,... = Multiplicative Fibonacci a(n)=a(n-1)*a(n-2): a(n-1) ^ phi is asymptotic to a(n)
2,2,4,16,65536,1.158*10^77....
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Might results which show the same result for tetration as for exponentiation be of any use (like in the range from 2 to e^(1/e))?
I experimented with this and found 9 numbers which have the same height and exponent and show nearly the same result for tetration and exponentiation.
Might this be of any use or worth looking at it ...
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Tetration and $f(x) = \exp(\int_1^x \ln(f(t)) dt)$
Let $g(x)$ satisfy $g(1) = 1 , g'(1) = 1 , g(1+x) = \exp(g(x))$
Now it is clear that $g'(5) = g(5)g(4)g(3)g(2)$
This invites to think of the function $f(x)$ which is defined similarly and might or ...
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Which hyperoperations produce a "prefix-complete" sequence?
Definition ("prefix-complete"): A sequence of positive integers $(a_n)_{n=1,2,3,\dots}$ will be called prefix-complete in base $b$ iff, for any positive integer $p$, there is some $a_n$ ...
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Sources on how tetration is defined in set theory and on infinite power towers
I was able to find the set theory definitions of addition and multiplication, but not of tetration. I wondered if somebody could define tetration in terms of set theory, and hopefully provide some (...
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Tetration Power Series
While reading through the Citizendium article on tetration, the first hyper-operation above exponentiation, I came across a power series approximate of tetration. The article said that it got the ...
Anthony Corsi