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.

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Inverse hyperoperation code library

I hope you're having a nice day, I am an Electrical Engineer and I am trying to do a math paper, but I need a coding library in any coding language that can do inverse hyperoperations, do you know of ...
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“Shooting Room” ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
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We know that $2+2 = 2 \times 2 = 2^2$. Is this true for all successive hyperoperations?

Given that multiplication is repeated addition and exponentiation is repeated multiplication, if we were to continue onwards, what would the result be for successive hyperoperations? If we take the ...
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“Multi-distributive” operations on sets.

As you can remember from Algebra courses, if $A$ is a set, and exist functions $\oplus , \odot: A \to A$ such that $(A,\oplus,\odot)$ is a ring, then such ring has a distributive property such that $$\...
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Using pentration with non-integer values: how to solve 2^^^1.5? [duplicate]

Say we define using hyperoperators with non-integers the same way we do with exponentiation, that is to say we convert the 'exponent' to a fraction and raise the base to the hyper-power of the ...
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Extend hyperoperation series to index in $\mathbb{R}$ [duplicate]

The hyperoperations are a series of binary operators that are created via iteration. The zeroth member in this series ("operation 0") is a simple increment , $a' = a+1$. Incrementing the number $x$ $...
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Tetration: Summation of $\displaystyle {1 \over{}^{n}2}$

I'm going to get straight to point with this question - Can you find a closed form solution to this sum. $$\sum_{n=1}^\infty \displaystyle {1 \over{}^{n}2}$$ (where ${}^{n}2$ represents the nth ...
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Are there more identity numbers past 2?

I apologize in advance if this question is too ill-posed, but here we go. As an additive identity, $x+0=x$. As a multiplicative identity, $x\times 1=x$. $2$ feels similar in a way I can't define as ...
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New commutative hyperoperator?

After reading about Ackermann functions , tetration and similar, I considered the commutative following hyperoperator ? $$ F(0,a,b) = a + b $$ $$ F(n,c,0) = F(n,0,c) = c $$ $$ F(n,a,b) = F(n-1,F(n,a-...
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Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
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Fractional and complex rank hyperoperations [duplicate]

The first hyperoperation $H_1(a, b) = a + b$ is addition, and is defined over the complex numbers. The second hyperoperation $H_2(a, b) = ab$ is multiplication, and is also defined over the complex ...
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Non integer Hyper-powers. [duplicate]

If I have a function $y=x^x$, that can be denoted in hyperpower notation as $^2x$, but I will be denoting it as $ y= $ hyp$_2(x) $. In general, for hyperpowers, $y=x^{x^{x^{...}x}}$ or in my notation $...
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Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is ...
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Tetrating by non-integers? [duplicate]

Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (x[4]x), which is the same as x pentated by 2 (x[5]2), would look like on a graph. To do this, ...
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A pre-successor hyperoperation: subplus

I wrote up these notes about a month ago and just found them. I could really use a second opinion or two. I notice there's a whole question about how my whole thesis is wrong, but that doesn't ...
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Does $2^{k!}$ have any significance w.r.t. primality or Wilson's Theorem?

Typically, Wilson's is given as $$(n-1)! \equiv -1 \pmod{n},$$ which is short and sweet, but I came up with an alternate presentation of it that's arguably needlessly complicated, but also arguably ...
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Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$ f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n). $$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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Why order matters in hyperoperations after exponentiation? [duplicate]

In first and second hyperoperations (addition and multiplication), order of two operands doesn't matter, result is same. However, it doesn't work on exponentiation. Why it does matter in ...
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Domain for Knuth's up-arrow

The wikipage for Knuth's up-arrow notation says $a \uparrow ^n b$ is defined recursively for integer $a$ and non-negative integers $b$ and $n$ like so: \begin{equation} a\uparrow^n b = \begin{cases}...
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How does this hyperoperator for sin^2 converge and what's the limit?

I was riding my bike yesterday and thinking about the following expression: Suppose $$S_3 = \sin^2(\frac\pi2) + \sin^2(\sin^2(\frac\pi2)) + \sin^2(\sin^2(\sin^2(\frac\pi2)))$$ which serves to define ...
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What Tetration Algorithm can I utilize

I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow": A mandelbrot fractal (I call it "operation order 1") is done by ...
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How to compute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $ \int{x↑↑n} dx $ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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A new notation for operations and some questions [duplicate]

let $x^{/1/}=x+x$ (addition) $x^{/2/}=x.x$ (multiplication) $x^{/3/}=x^x$ (exponentiation) $x^{/4/}={}^xx$ (tetration) and so on..... I have the following questions: $(1)$ Can we define an ...
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Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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Extending Knuth up-arrow/hyperoperations to non-positive values [duplicate]

So... I had the silly idea to extend Knuth's up-arrow notation so that it included zero and negative arrows. It is normally defined as $$\begin{align*} a \uparrow b & = a^b \\ a \uparrow^n b & ...
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Why do we have the present order of operations, and how do hyperoperations fit in?

Something that's been bugging me for a fairly decent while is the order of operations - not so much using it, however, as to understanding where it comes from. Typically we're introduced to it in the ...
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What is the name for this: “x^^y” (as in 2^^2=2^2 and 3^^3=3^3^3 and so on)

How do you call / is there any specific name for the following: x^^y e.g.: 2^^2 = 2^2 3^^3 = 3^3^3 4^^4 = 4^4^4^4 ...
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Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher ...
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Tetration by a Non-Integer

Does anyone think that tetration by a non-integer will ever be defined ... really properly? Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time ...
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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. $...
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Could there be an $\omega_1^{CK}$th hyperoperation? [closed]

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 ...
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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 ...
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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 ...
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How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?

3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms ...
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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^{\...
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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 ...
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Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed? [duplicate]

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)),&...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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82 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 ...
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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....
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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?
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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 $n\ge 3$ ?...
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152 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 ...
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87 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 ...
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340 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 ...
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138 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 ...