Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [hyperoperation]

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

0
votes
0answers
8 views

Can we evaluate noninteger hyperoparations?

The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on. What happens when $n$ is noninteger? Can we evaluate, e.g. $...
-1
votes
1answer
51 views

Could there be an $\omega_1^{CK}$th hyperoperation?

If addition is the first hyperoperation, multiplication is the second, and the $(\alpha+1)$th hyperoperation is repeated occurrences of the $\alpha$th one. Is it possible for a limit ordinal (for ...
1
vote
0answers
12 views

Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
0
votes
0answers
29 views

Can there be fractional hyper operators and if so how to they function? [duplicate]

If we call addition (and subtraction as they are the same) the first hyperoperator, multiplication (and division) the second, exponentiation (radicals, indices and logarithms) the third, then ...
2
votes
1answer
93 views

Question about $TREE(3)$ and Graham's Number

Lets say: $G=\text{Graham's Number}$. And: $$ \begin{align*} \alpha_1&=G\uparrow^G G, \\ \alpha_2&=\alpha_1\uparrow^{\alpha_1} \alpha_1 \\ &\vdots\\ \beta_1 &= \alpha_G\uparrow^{\...
2
votes
1answer
29 views

Hyperoperation: Why does $H_n(0,b) = 0$ for $n\ge4$, $b$ odd ($\ge -1$)

On the Wikipedia for hyperoperations is says that $H_n(0,b) = 0$ if $n\ge4$, $b$ odd ($\ge -1$). From what I gether, the basic jist of what this is saying is that $0^{0^0}=0$ and from that we can ...
0
votes
0answers
40 views

Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed?

Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper): $$ \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...
0
votes
2answers
79 views

There is a way to write TREE(3) via $F^a(n)$?

I read about Graham number and TREE(3). Graham number is: $f^{64}(4)$ where $f(n)=3\uparrow^n 3$ My question is: If there is a way to write ...
3
votes
2answers
116 views

Where does this array-based fast-growing function fall in the fast-growing hierarchy, and how does it compare to TREE(n)?

[See below for a clarification edit and progress thus far] So I have been reading into the "fast-growing hierarchy" of functions, and I devised this (somewhat convoluted) function for generating very ...
0
votes
2answers
42 views

Are hyperoperations < 3 to a reciprocal of a positive integer equivalent to the 'root' inverse to that integer?

Can the logic of $\sqrt[n]x = x^{1/n}$ be applied to tetration and other natural numbered hyperoperations greater than exponentiation, or, do reciprocals of positive integers as the second argument of ...
1
vote
2answers
83 views

Is there any function that like this function?

I got a idea from fast-growing hierarchy function to create new function g.(I think it is computable.) $$g_0(n) = n + 1$$ $$g_{a+1}(n) = g_a^{g_a(n)}(n)$$ Which different from fast-growing ...
0
votes
0answers
16 views

Is tangent space a separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane?

Entire question should be: Can we consider tangent space as the separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane? Is possible to ...
0
votes
0answers
27 views

Generalized algebraic operations

I've looked around the web alittle, and besides hyper-operations it seems there has been little to no attempts at generalizing binary operations. Is there an operation space, where the operators exist....
1
vote
1answer
64 views

Hypersubstitution, m-ary terms, semigroups, equivalent definitions

A hypersubstitution $\sigma$ is (see, for example, Universal Algebra and Applications in Theoretical Computer Science, by Denecke and Wismath) mapping from term $f_i(x_1,...,x_{n_i})$ to the term $\...
1
vote
1answer
67 views

Hyperidentity, semigroups, bands.

Let a semigroup satisfy $F(x,x)\approx x$, where $F$ is a binary operation symbol.Let $B$ satisfy $x(yz)\approx (xy)z$ and $xx\approx x$. Does $B$ satisfy $F$ as a hyperidentity?We need only consider ...
1
vote
1answer
61 views

Semigorup variety, hyperassociativity,idempotentunclear proof of $x^4\approx x^2$

Let $V$ be a hyperasociative semigroup variety. For hyperasociativiy see below. Then $V$ satisfies the following identity: $$x^2 \approx x^4.$$ A proof attempt is given here: If $V$ is idempotent (i.e....
2
votes
0answers
62 views

How to solve $ax^x+bx+c=0$?

How can I solve $$ax^x+bx+c=0$$ or $$ax^{x^x}+bx^x+cx+d=0$$ where $x^x$ and $x^{x^x}$ - tetration? Is there analogue of discriminant for it?
2
votes
2answers
117 views

Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven?

See here : http://googology.wikia.com/wiki/Arrow_notation for the definition of the up-arrow function. Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven for all $m\ge 1$ and $n\ge 3$ ?...
2
votes
0answers
88 views

Finite algebraic structures where all hyperoperations (addition, multiplication, exponentiation, tetration, etc.) are well-defined

Let $\langle R, +, \times, \uparrow, \uparrow\uparrow, \uparrow\uparrow\uparrow, \ldots; 0, 1\rangle$ be an algebraic structure with two constants $0, 1$ and where an infinite sequence of binary ...
1
vote
1answer
72 views

Hyperidentities and Clones, Trivial observation, commutativity

In the book Hyperidentities and Clones they (Denecke and Wismath) write: $xy \approx yx$, in other words $$F(x,y)=F(y,x)$$ considered as a hyperidentity implies $$x\approx y.$$ I would like to ...
0
votes
2answers
98 views

3↑↑↑3= ? but with 10 instead of 3 ( approximation, order of magnitude )

3↑↑↑3= (or near) in power tower of 10 or in ( Knuth ) arrow ↑ notation of 10 to get a sense of it's order of magnitude; I grasp numbers more easily with 10 3↑↑↑3 being the first really huge number in ...
1
vote
1answer
128 views

Better bounds on my ordinal hyperoperators

I've defined some ordinal hyperoperators as follows: $$a\{b\}c=\begin{cases}a+1,&b=0{\rm~or}~c=0\\\sup\{(a\{b\}x)\{y\}a:x<c,y<b\},&\text{else}\end{cases}$$ For $a$ and $c$ that satisfy ...
12
votes
4answers
599 views

Can tetration 'escape' the complex plane?

Considering subtraction can break out of the natural numbers and into integers, division can break out of integers and into rational numbers, and exponentiation can break out of rational numbers and ...
0
votes
0answers
135 views

How are hyperoperations of rational and irrational numbers calculated?

Let's say I have the following problem, where x represents a real number. x = 2 * 3.4 I'm sure there are many ways one could solve this problem, but the way I would solve this is by adding 3.4 to 3....
5
votes
2answers
339 views

Pentation Notation - How does it work? [duplicate]

When going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^3}}=...
5
votes
3answers
467 views

Is there a notation for the repetition of basic operations?

I mean... multiplication is the repetition of addition: $2*2 = 2+2$ $3*3 = 3+3+3$ exponential is the repetition of multiplication: $2^2 = 2 * 2$ $3^3 = 3*3*3$ .. it is an obvious pattern. I ...
4
votes
1answer
190 views

Tighter bounds on the fast growing hierarchy?

Not a dupe of this question, as I'm searching for tighter bounds. We define the fast growing hierarchy for finite values as follows: $$f_k(n)=\begin{cases}n+1,&k=0\\f_{k-1}^n(n),&k>0\end{...
3
votes
0answers
77 views

For which n does the nth “hyperoperation number” n[n]n begin with n in base 2?

If $[n]$ denotes the $n$th binary hyperoperation in the sequence $(+,\times,\uparrow,\uparrow\uparrow,...)$, then the following equality is readily verified for $n=1,2,3,4:$ $$n\,[n]\,n\ =\ (n_2...)...
8
votes
1answer
189 views

Are there nontrivial equations for hyperoperations above exponentiation?

A similar question was asked in comments elsewhere. A paper by Roberto Di Cosmo and Thomas Dufour ("The Equational Theory of 〈ℕ, 0, 1, + , ×, ↑〉 Is Decidable, but Not Finitely Axiomatisable") asserts ...
6
votes
2answers
475 views

Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
1
vote
0answers
171 views

How to extend this extension of tetration? [closed]

if $0\le b<1$, then $a↑↑b = a^b$ if $b\ge1$, then $a↑↑b = a^{a↑↑(b-1)}$ if $b<0$, then $a↑↑b = \log_a(a↑↑(b+1))$ so for example, $2↑↑\pi = 2^{2^{2^{2^{0.1415926...}}}} = 21.5963561$ How can ...
14
votes
2answers
1k views

How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\...
0
votes
1answer
152 views

Derivative of super square root [closed]

What is the derivative of $y=^{1/2}x$? I tried finding the derivative of $x^{x}$ and then finding the inverse of that, but that didn't work.
2
votes
1answer
138 views

Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations?

Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the ...
1
vote
0answers
22 views

Recursive function defining basic addition [duplicate]

EDIT: There is another question, just like this one, by "Bert van den Bosch", if you wish to see another thread with some detailed answers to this question. They may have some better explanations to ...
21
votes
5answers
1k views

what operation repeated $n$ times results in the addition operator?

I had a difficult time in phrasing my question. But I was wondering if there is an operation that, when repeated n times, results in the addition operator. Same way as repeating addition n times ...
3
votes
2answers
143 views

Is this a new operation? [duplicate]

I was thinking "What is before addition", and came up with this. This analogy describes it: Addition is to multiplication as [operation] is to addition. On wikipedia it says it is just "1+b". I came ...
16
votes
2answers
837 views

Could someone tell me how large this number is?

Context: If you guys didn't know, I'm running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to ...
0
votes
2answers
302 views

Why is tetration considered the next step after exponentiation? [closed]

Tetration is often stated to be the next step after exponentiation (see for example Wikipedia): $$\large a^{a^{a^{...^a}}}$$ Where the exponents are taken $b$ times from the top. I refer to the ...
1
vote
1answer
119 views

What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$?

What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$? I know we can find the least significant digits in constant time using modular arithmetic, is the most ...
2
votes
0answers
141 views

Are 'numerals' closed under exponentiation?

I have read Edward Nelson's Warning signs of a possible collapse of contemporary mathematics a couple of times, it is a very interesting read, but I do not understand the conclusory paragraph. In ...
2
votes
1answer
98 views

Is there a mathematical term, practical application, or area of math that covers a function raised to itself?

Some abstract examples would be: $f(x)^{f(x)}$ or $f(x)^{f(x)^{f(x)...}}$ Actual equations I've attempted to look at can be viewed here on desmos.com There seems to be a pattern of common convergence, ...
5
votes
2answers
161 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{ ...
2
votes
2answers
133 views

What can we say about the prime factors of $​^{10}10+23$?

In a video on ultrafinitism I saw a claim that the number $​^{10}10+23$ does not have prime factorization. While I don't accept the premise of ultrafinitism, I got curious, what can we say about the ...
2
votes
1answer
228 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 ...
1
vote
0answers
111 views

How do hyperoperations like tetration exist if operations are seperate relations and not repeatitions of each other.

I've run into a bit of a conflict in my fundamental understanding of concepts in math. I've always known the arithmetic operation to be extensions of each other. Multiplication is repeated addition, ...
2
votes
2answers
389 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?
0
votes
1answer
63 views

Is $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)) = H_n(\displaystyle\lim_{h \to 0} f(h), \displaystyle\lim_{h \to 0} g(h))$ true for all $n$?

Consider the limit $\displaystyle\lim_{h \to 0} H_n(f(h), g(h)), $ where $H_n(a, b)$ denotes the $n$th hyperoperation $H_n(a,b) = a \uparrow^{n-2}b$ with both $f(x)$ and $g(x)$ being continuous and ...
1
vote
2answers
60 views

Domain of the n composed logarithms on x.

As we know, the real logarithm has the domain $$ D_1 = \{x : x \in \mathbb{R}, x > 0\} $$ What is the logarithmic domain of "higher order" logarithms, at index n? For example, it seems that $$...
1
vote
1answer
479 views

What is the inverse to hyperoperation for positive integers?

According to Wikipedia hyperoperation for positive integers is defined as $$ H_{n}(a,b)=H_{n-1}(a,H_{n}(a,b-1)) $$ with some base conditions. (We take $ n \geqslant 1 $.) Question: Recursivly define ...