# 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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### 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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### 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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### 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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### What about series involving strong primes?

I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
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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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### 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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### 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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### 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 ...
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 ...
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 ...