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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2answers
301 views

Tetration graph for this function

I'm trying to visualize the derivatives of exponential tetration, by taking the original equation from its graph. Right now there is no elementary way of expressing the derivative. I'm not allowed to ...
2
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1answer
77 views

A question on the equation $^qx=2$

Given the equation $$^qx=2$$ with $q\gt3$ where $^qx$ means the 'tetration' operation on $x$, my question is: is it possible to find a value for $q$ for which the solution $x$ of the equation is a ...
3
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0answers
198 views

Efficient algorithm for calculating the tetration of two numbers mod n?

I'm trying to study the algebraic properties of the magma created by defining the binary operation $x*y$ to be: $ x*y = (x \uparrow y) \bmod n $ where $ \uparrow $ is the symbol for tetration. ...
2
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1answer
109 views

Is the exact value of $^{\infty}i$ known? [duplicate]

Wikipedia's article on tetration has a table of successive tetrations of $i$ that seems to imply that $^{\infty}i$ converges, and my own experimentation seems to confirm it, but I'm suspicious of ...
10
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4answers
350 views

If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$?

In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. $y=\sqrt{6+\sqrt{6+\...
5
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1answer
216 views

Does infinite tetration of negative numbers converge for any value other than -1?

Okay, so I know that for positive values, $^{\infty}x$ converges to $-\frac{W(-\ln x)}{\ln x}$ for $e^{-e}\le x \le e^{\frac1e}$. Above that, it diverges. For positive values less than $e^{-e}$, any ...
1
vote
1answer
394 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
4
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2answers
119 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
6
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2answers
247 views

Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, … don't change, and generalisation

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
4
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0answers
136 views

Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
4
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0answers
135 views

Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...
4
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0answers
481 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
11
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1answer
166 views

Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
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1answer
165 views

Comparison between two tetrations

For a given natural number n, what ist the least number m, such that $$e \uparrow \uparrow m > \pi \uparrow \uparrow n$$ It seems that m = n + 1 is the desired number. Is this true for all n ?
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0answers
138 views

Is there a proof that the number $2 \uparrow \uparrow m+3 \uparrow \uparrow n$ is always squarefree?

I searched prime factors of the numbers $$z(m,n) := 2 \uparrow \uparrow m + 3 \uparrow \uparrow n$$ where $m,n\ge1$ Interestingly, z(3,3) is prime, the largest prime I found so far and probably the ...
4
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1answer
176 views

Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
1
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1answer
76 views

Does $a \uparrow \uparrow (n+1)-a \uparrow \uparrow n$ divide $a \uparrow \uparrow(n+2) - a \uparrow \uparrow(n+1 )$?

Does $$a\uparrow \uparrow (n+1) - a\uparrow \uparrow n$$ divide $$a\uparrow \uparrow (n+2) - a\uparrow \uparrow (n+1)$$ for all $a,n \ge2$ ? The case n = 0 is easy : $$a^a-a=a(a^{a-1}-1)$$ and $...
1
vote
0answers
161 views

Question about tetration modulus a prime $p>100$

Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$. See : http://en.wikipedia.org/wiki/Tetration Let $p$ be a prime larger than $...
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1answer
176 views

Stationary prime factor 34276387 , confirmation and generalization wanted.

Sheldon L found out, that $$34276387$$ is a prime factor of $$2 \uparrow \uparrow n + 3 \uparrow \uparrow n$$ for any natural number $n \ge 5$. $2 \uparrow \uparrow n$ leaves the remainder ...
0
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1answer
193 views

How do we know that tetration is exclusively right-associative?

When we go from multiplication to exponentiation we lose commutativity ($3^2 \neq 2^3$). Perhaps when we go from exponentiation to tetration every operation yields two possible results.
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2answers
328 views

Calculating $a^n\pmod m$ in the general case

It is well known, that $$a^{\phi(m)}\equiv1\pmod m ,$$ if $\gcd(a,m)=1.$ So, $a^n\pmod m$ can be calculated by reducing n modulo $\phi(m)$. But, for the tetration modulo $m$ $$a \uparrow \...
14
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0answers
527 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
5
votes
2answers
263 views

radius of convergence of half iterate of sinh(z)?

The half iterate of sinh(z) has a formal power series, centered around z=0. Does the formal power series for the half iterate converge at the origin? This is equivalent to asking if the half iterate ...
1
vote
1answer
138 views

Slight confusion about the calculation of the last digits of graham's number

In Wikipedia, Graham's number, it is described how to calculate the last d digits of Graham's number. They introduce an algorithm simply iterating $$x = 3^x \mod 10^d$$ d times starting with x=3. ...
3
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0answers
220 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
1
vote
0answers
51 views

Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
24
votes
2answers
861 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...
4
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2answers
181 views

Which precision would be needed?

According to Wikipedia, it is not known whether the number $$\pi \uparrow \uparrow 4$$ is an integer. (See Tetration) To which precision would $\pi$ have to be calculated to decide this ? The ...
24
votes
1answer
1k views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
3
votes
2answers
225 views

Knuth's arrow up notation again

Consider the following recursion : $$ a(1)=3! $$ $$ a(n+1) = a(n)! \quad\hbox{for all $n\geq 1$} $$ For a given $n$, how can the number $m$ with $$ 10 \uparrow \uparrow m < a(n) < 10 \uparrow ...
4
votes
1answer
316 views

Knuth's up arrow notation

Given n, how can the number m with $10 \uparrow \uparrow m < 2 \uparrow \uparrow n < 10 \uparrow \uparrow (m+1)$ be calculated ? With induction, I got $10 \uparrow \uparrow m < 2 \...
1
vote
2answers
80 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x \...
0
votes
1answer
168 views

Does this follow from the definition of the LambertW function?

The LambertW function $W(s)$ also called ProductLog seems to satisfy this relation: $$-W(s) = \underbrace{-s e^{-s e^{\cdot^{\cdot^{-s e^{-W(s)}}}}}}_n$$ Or truncated: $$-W(s) = -se^{-s e^{-s e^{-s ...
4
votes
0answers
163 views

Arrow notation and decimals.

We know how to add with decimals. We know how to multiply with decimals. We know how to exponentiate with decimals. Do we know how to work with decimals for power towers? for example, can we deal ...
4
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1answer
404 views
2
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1answer
203 views

Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
2
votes
0answers
33 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton $X$...
3
votes
0answers
128 views

Does $^{\frac12}x=e^{W(\ln x)}$, or not?

Whenever I see tetration discussed here, I inevitably see it asserted that there's no consistent continuous definition for tetration. However, it seems to me that If we restrict ourselves to ...
10
votes
2answers
712 views

Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
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vote
2answers
279 views

Notation for function $ + \rightarrow \times $

Is there a standard notation to represent the function building the multiplication from the addition (I'm talking of the usual $+$ on $\Bbb N$)? I'm tempted by: $$ x \times y = + ^ y (x) $$ With ...
10
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1answer
148 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
1
vote
1answer
268 views

Fixed Point of $x_{n+1}=i^{x_n}$

For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$. EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$ My ...
104
votes
1answer
21k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not ...
9
votes
3answers
2k views

Calculating 7^7^7^7^7^7^7 mod 100

What is $$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$ I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
4
votes
1answer
103 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for $n\in\...
5
votes
1answer
246 views

Tetration of Alephs

Does this even mean anything? $\underbrace{x^{x^{x^{...^{x^x}}}}}_n$ Where $n = \aleph_0$? Because I "know" it converges when (say) $x = .5$ and $n \to \infty$
5
votes
4answers
350 views

Tetration limit

For each $n$, define $f_n:\mathbb R^+\rightarrow \mathbb R^+$ by $f_n(x) = \underbrace{x^{x^{x^{...^{x^x}}}}}_n$ Is it true that $\lim\limits_{n \to \infty} f_n(\frac{n+1}{n}) = 1$ ? A few ...
2
votes
1answer
113 views

Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?

I was trying to solve the recurrence $f_n=\exp(f_{n-1})$. My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$. The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$. if $f(m)...
2
votes
1answer
214 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to "...
25
votes
4answers
2k views

Seems that I just proved $2=4$.

Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$. Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$. Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and ...