Questions tagged [hyperoperation]

Question 1

When b is a real number non inclusively between 1 and $e^{e^{-1}}$, $b^x$ has two real fixed points. If b is increased to $e^{e^{-1}}$, the two real fixed points combine into one real fixed point. If ...

Question 2

Mathematicians use ∑ for repeated addition and ∏ for repeated multiplication. I’ve been exploring whether we can generalize this pattern for higher hyperoperations — such as exponentiation, tetration, ...

Question 3

One time, when I was just sitting around, I had a realization. A commonly known fact is that forward differences are a discrete analog of derivatives, and sums are a discrete analog of integrals. But, ...

Question 4

I was messing around with triangle numbers and commutative hyperoperators. Consider the zeroth commutative hyperoperator $F_0(a,b) = \ln(e^a + e^b)$. The "zero-order" triangle numbers would ...

Question 5

There is a hierarchy of commutative hyperoperators, which, in addition to commuting, distribute over the previous operator in the hierarchy. The zeroth operatorion this list is $F_0(x, y) := \ln(e^x + ...

Question 6

It has been asked if there is an operation, that when repeated $n$ times results in addition. The solution given is that one can use the $\max$ function in a clever way: $$\max(a, b) + 1 + \delta_{ab} ...

Question 7

I beginning to start math from the start again and I'm trying to understand things more intuitively and I'd like some help to understand associativity. I can understand it as the characteristic of an ...

Question 8

There exists a series of commutative hyperoperations with the defining trait that each operation is distributive over its predecessor. The operations are defined recursively, taking $$a \times_{n} b = ...

Question 9

I’m an independent learner without formal training in mathematics. While studying base-2 tetration, I observed a localized logarithmic identity that holds from $A_3$ to $A_5$, and breaks at $A_6$. I’d ...

Question 10

Prime numbers can defined as numbers $p$ for which there are no $a$ and $b$ besides $1$ and $p$ such that $p = a\cdot b$. If we define $\pi(n)$ to be the number of primes less than or equal to $n$, ...

Question 11

Tetration is defined as “repeated exponentiation” - that is, $2$ tetrated to $5$ is equal to $2^{2^{2^{2^2}}}$, just as $2$ exponentiated to $5$ is equal to $2\cdot{2}\cdot{2}\cdot{2}\cdot{2}$ (...

Question 12

Let $n > 1$ be an integer not a multiple of $10$. Is there a short proof that $n = 5$ is the only solution to $n^n \equiv n^{n^n} \pmod {10^{n-1}}$, given the fact that $5^{5^5} \equiv 3125 \pmod{...

Question 13

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 $a_n$ ...

Question 14

While I was submitting a few sequences to the OEIS, I noticed an asymmetrical pattern involving the rightmost digits of an interesting set of well-known integer sequences. Let $a \in \mathbb{Z}^+$, $n ...

Question 15

We all know that many years ago we invented powers. e.g. $3^4$ meant how many times we multiply 3. i.e. $3^4=3\cdot 3\cdot 3\cdot 3$. But then, people started asking questions like what is the ...

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Questions tagged [hyperoperation]

Pentation from three fixed points

Proposal for a symbol for hyperoperation aggregation [closed]

Deriving Formulas for Hyperoperational Derivatives

Does $n - \ln\left(\sum_{i=0}^{n}\exp(i)\right)$ converge to approximately $-0.458675145$?

Are there any elementary operations which behave like $a^{\ln(b)}$?

Is there an operation when repeated $n$ times on a number $x$ gives $\max(x, n)$?

What are the semantical pre-requisites for an operation being associative?

How to prove that each commutative hyperation operation is distributive over the previous?

Discovered Local Logarithmic Identity in Base-2 Tetration ($A_3$ to $A_5$) Before Structural Breakdown at $A_6$ – Request for Validation

Is there an asymptotic for so-called "exponential primes"?

Does tetration go up or down? (and other related questions) [duplicate]

Easy proof that $n=5$ is the only solution of $n^n \equiv n^{n^n} \pmod {10^{n-1}}$ if $n \in \mathbb{N}-\{0,1\}$ is not a multiple of $10$

Which hyperoperations produce a "prefix-complete" sequence?

Final digit of $a[n]a$, where $[n]$ is the $n$-th hyperoperator and $a \in \mathbb{Z}^+$

Definition of tetration [duplicate]

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