# Questions tagged [prime-constellations]

On certain subsets of prime numbers which are consecutive and close. Prime twins p and p+2, as well as p-2,p,p+4, are constellations. Also related are admissible sets in number theory, which are sets A of integers a_i such that there may be an integer t with many or all of t+a_i being prime. This has ties to prime gaps and additive number theory

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### The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$

A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted gn or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...

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### Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?

Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,...

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### Remainder-balancedness of primes

Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\...

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### Can the twin-prime conjecture be related to the growth of $\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)$?

This question is based on this Math StackExchange question by user buja and my related Math StackExchange question which was asked about a month ago and is still unanswered.
I believe
$$\phi_2(n)=n \...

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### Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?

Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...

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### Geometric prime distribution

Let integers $\ a>1\ $ and $\ b\in\mathbb Z\ $ be relatively prime (hence $\ b\ne 0).\ $ The Dirichlet's prime distribution theorems apply to the arithmetic sequence
$$ (_aG_b(x) : x\in\mathbb Z) $$...

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### Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?

Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...

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### Prime constellations equivalent up to permutation

This question generalizes Symmetry in Hardy-Littlewood k-tuple conjecture. Say two prime constellations are equivalent up to permutation if they consist of the same multiset of prime gaps. One can ...

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### Measuring philoprimality/misoprimality

Given a natural integer $x$, let
$$\alpha(x)=(\log x)^2\sum_{p\in\mathcal P\setminus\{x\}}\frac{1}{(x-p)^2}$$
(with $\mathcal P$ denoting the set of prime-numbers)
measure its "philoprimality&...

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### Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$?

猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如
24 25 26 27 (2 3 5 13)
其余长度为4的连续合数序列要么素因子个数大于4 要么素因子集合不等于{2 3 5 13} 这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌，非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。
...

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### Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...

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### $t$-balanced numbers

Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...

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### On a conjecture about the arithmetic function that counts the number of twin primes

This is cross-posted from the question that I've asked with same title on Mathematics Stack Exchange two months ago, which has remained unanswered.
Given a positive real number $x$ we will write ...

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### Is this conjecture equivalent to Polignac's conjecture?

Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...

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### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes

I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...

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### Asymptotic behaviour of $\sum_{k=1}^{n}\frac{R_{k+1}+R_k}{R_{k+1}-R_k}$, where $R_k$ are the Ramanujan primes

I was inspired in the following post from this Math Overflow that was asked yerterday Asymptotic behavior of a certain sum of ratios of consecutives primes to ask about a similar question for ratios ...

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### Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?

I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^...

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### On the quantity of twin prime pairs of a given form

Let $p_l$ the $l$-th prime number. I've considered the formula
$$\frac{N_{n+1}}{N_n}+\frac{N_{n+2}}{p_{n+1}N_n}\pm1$$
where $N_k=\prod_{l=1}^k p_l$ is the primorial of order $k$. Previous formula ...

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### Is this theorem on the abundance of prime patterns/k-tuples known?

I am looking for references regarding the following statement.
For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for ...

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### A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture

I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture
$$\sum_{\substack{\text{...

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### On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture

I wondered, inspired in a result from [1] (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum_{\substack{\text{...

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### What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?

It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...

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### Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...

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### Sergei numbers : even integers n being a prime gap at least n times

Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...

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### Can this weakening of Polignac's conjecture be proven?

Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...

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### Are the elements in the n-th row of the first matrix a permutation of the elements in the n-th row of the second matrix?

From my previous questions here and here the following two matrices arise for twin primes and cousin primes from Dirichlet convolution:
For $h=2$ twin primes:
$$T_2(n,m)=\sum\limits_{\substack{k=1 \\...

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### Euclides' sieve

This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, ...

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### Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?

In this previous question of mine
I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture ...

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### Admissible k-tuples and primorials

Let $ (a_{1},\cdots,a_{k}) $ an admissible $ k $ -tuple and $ P_{k} $ the product of the first $ k $ primes. Do we have a conjectural expression for the number of positive integers $ n $ not ...

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### Do prime gaps that are a power of "h" have the same density?

Send me back to Mathematics Stack Exchange if this question is not research level.
At Terence Tao's blog post there is the expression:
$$\sum\limits_{n \leq X} \Lambda(n)\Lambda(n+h) \ \ \ \ \ \ \ \ \ ...

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### On a coprime generalization of Cramer's conjecture

Given a large enough integer $n\in\Bbb N$ and a real $r\in\big(0,\frac12\big]$ and $n_1\in\Bbb N_{> n}$ is the smallest integer such that $n_1=AB$ for two coprime integers $A$ bigger than but close ...

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### What is the narrowest interval I=[a,b] such that there are infinitely prime gaps of size in I?

Polymath8b project allowed, building on Zhang's 2013 breakthrough, to prove that there are infinitely prime gaps of size less or equal to 600. Under the generalized Elliott-Halberstam conjecture, one ...

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### Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions:
$$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$
$$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$.
Let's ...

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### Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...

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### Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2) $?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.
Define
$$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$
Is it consistent with current ...

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### Counting function for prime pair with bounded gaps between them [duplicate]

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible.
Yitang Zhang breakthrough result established that ...

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### Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used?
In their paper,
Some problems of 'Partitio numerorum'; III - On ...

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### Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...

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### Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly $\dfrac{x}{\...

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### On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...