If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...

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15
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2answers
516 views

What is $i$ exponentiated to itself $i$ times?

I was just wondering about this. I searched about it on the net and found that it is called tetration and after this comes pentation and then hexation and so on so forth. I don't really understand ...
1
vote
1answer
240 views

Find $N$, in the decimal expansion of the large number $N=4^{4^{4^4}}$

Find $N$, in the decimal expansion of the large number $$N=4^{4^{4^4}}$$ Following on from the question I posted yesterday about finding the number of digits ( Find the number of digits, $D$, in ...
1
vote
3answers
458 views

Find the number of digits, $D$, in the decimal expansion of the large number $N=4^{4^{4^{4}}}$

The full question is: Find the number of digits, $D$, in the decimal expansion of the large number $$N=4^{4^{4^{4}}}$$ Try and find the most efficient ways of finding $D$. I know that ...
5
votes
0answers
247 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{^} ...
4
votes
1answer
155 views

Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
11
votes
4answers
1k views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
11
votes
9answers
2k views

half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and ...
5
votes
1answer
291 views

Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration: $$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$ And now I am ...
2
votes
2answers
343 views

What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
2
votes
4answers
298 views

a tetration limit for base $a > e^{1/e}$

Let $a$ be a real number with $a > e^{1/e}$ and $a <> e$. $slog$ means superlog base $e$ and $sexp$ means superexp base $e$. $sloga$ means superlog with base $a$ and $sexpa$ means superexp ...
1
vote
5answers
198 views

Limit of a recursively defined bivariate function.

Let m and n be positive integers. Let $f(m,0)=m$ Let $f(m,n)= e \ln(f(m,n-1))$ $$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$ Where $C$ is a constant. It seems ...
13
votes
2answers
540 views

A puzzle with powers and tetration mod n

A friend recently asked me if I could solve these three problems: (a) Prove that the sequence $ 1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of ...
26
votes
1answer
384 views

Iterated exponent of $i$

WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
4
votes
1answer
498 views

Calculating the residue of power towers

I want to calculate the residue of a power tower. How do I do that? For example, I want to know the answer to this: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
10
votes
2answers
532 views

Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
1
vote
2answers
214 views

Find the remainder in the following case where there's a infinite power tower of $7$.

What is the remainder when $$7^{7^{7^{7^{.^{.^{.^{\infty}}}}}}}$$ is divided by 13? I'm getting $6$. Is it correct?
5
votes
2answers
629 views

Last few digits of $n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}$

I want to compute last few digts (as much as possible ) of the following number $$ N:=n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}\!\!\!\hspace{5 mm}\mbox{ if there are $k$ many $n$'s in the expression and ...
12
votes
2answers
570 views

Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function? ...
5
votes
1answer
454 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
8
votes
0answers
206 views

Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]

Possible Duplicate: Infinite tetration, convergence radius Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that ...
2
votes
3answers
181 views

Mathematical function for the powers

I have this formula $$\underbrace{2^{2^{2^{.^{.^{.^{2^2}}}}}}}_n$$i.e. where the total number of 2's is $n$. Is there any way to write it as a single mathematical function?
21
votes
1answer
2k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...
6
votes
0answers
329 views

Is ${^5\pi}$ an integer? [duplicate]

Possible Duplicate: How to show $e^{e^{e^{79}}}$ is not an integer Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it? Here ${^5\pi}$ means the result of tetration ...
59
votes
4answers
4k views

Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?

Problem: Find $x$ in $$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$ Trick: $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so, $x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and, ...
8
votes
3answers
707 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
23
votes
2answers
2k views

Is there a natural way to extend repeated exponentiation beyond integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
4
votes
0answers
266 views

What is the Equivalent Form of Tetration to the Exponential $n^{1/n}$?

I've been working on a project for a wiki that I'm a member of. It is the Sequence of the Day for September 2. You can see my progress at ...
41
votes
11answers
3k views

Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
2
votes
1answer
264 views

Could real iterates of the Taylor Series expansion of $b^x$ help to find a way to define tetration?

When we consider the Taylor Series expansion of $f(x)=b^x$ for some $b \in \mathbb{R}$, we see that $$b^x = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}x^n.$$ We can substitute $x$ for $b^x$ to find ...