# 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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### 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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### 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 ...
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 $\pi(n+r_{0}(n))-\pi(n-... 1answer 219 views ### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ... 0answers 240 views ### A conditional approach to twin prime conjecture Disclaimer: this question was first asked on a French forum (here comes the link for those of you who read French: http://www.les-mathematiques.net/phorum/read.php?5,1758830,1758922), but no ... 0answers 75 views ### Asymptotic behaviour of$\sum_{k=1}^{n}\frac{R_{k+1}+R_k}{R_{k+1}-R_k}$, where$R_k$are the Ramanujan primes I was inspired in the following post from this Math Overflow that was asked yerterday Asymptotic behavior of a certain sum of ratios of consecutives primes to ask about a similar question for ratios ... 1answer 239 views ### Are there infinitely many primes of the form$\frac{3a^2-a}{2}+b^4$? I was inspired from a theorem due to Iwaniec and Friedlander, see , to ask the following conjecuture involving integers. Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^... 1answer 133 views ### On the quantity of twin prime pairs of a given form Let p_l the l-th prime number. I've considered the formula$$\frac{N_{n+1}}{N_n}+\frac{N_{n+2}}{p_{n+1}N_n}\pm1$$where N_k=\prod_{l=1}^k p_l is the primorial of order k. Previous formula ... 1answer 193 views ### Is this theorem on the abundance of prime patterns/k-tuples known? I am looking for references regarding the following statement. For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for ... 1answer 170 views ### A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture I wondered, inspired in a result mentioned from  (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture$$\sum_{\substack{\text{... 1answer 155 views ### 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  (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture, $$\sum_{\substack{\text{... 0answers 121 views ### What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes? It is known that the twin prime conjecture is a special case of the k-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld. On the ... 1answer 872 views ### Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series,$$\Lambda(m)=\... 1answer 138 views ### 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 ... 1answer 229 views ### 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 $such that the ... 0answers 94 views ### Are the elements in the n-th row of the first matrix a permutation of the elements in the n-th row of the second matrix? From my previous questions here and here the following two matrices arise for twin primes and cousin primes from Dirichlet convolution: For$h=2$twin primes: $$T_2(n,m)=\sum\limits_{\substack{k=1 \\... 0answers 663 views ### Euclides' sieve This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products P_i of the elements in the set, ... 0answers 137 views ### Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture? In this previous question of mine I introduce under Goldbach's conjecture the notation r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} as well as the related so-called NFPR conjecture ... 1answer 198 views ### Admissible k-tuples and primorials Let (a_{1},\cdots,a_{k}) an admissible k -tuple and P_{k} the product of the first k primes. Do we have a conjectural expression for the number of positive integers n not ... 1answer 441 views ### Do prime gaps that are a power of "h" have the same density? Send me back to mathematics stack exchange if this question is not research level. At Terence Tao's blog post there is the expression:$$\sum\limits_{n \leq X} \Lambda(n)\Lambda(n+h) \ \ \ \ \ \ \ \ ... 2answers 292 views ### 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 ... 1answer 167 views ### 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 ... 0answers 169 views ### 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 ... 1answer 303 views ### 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 ... 0answers 176 views ### 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 ... 1answer 251 views ### 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 ... 0answers 287 views ### 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 ... 3answers 396 views ### 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 ... 2answers 467 views ### 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}{\...
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 ...