All Questions
25 questions
8
votes
2
answers
393
views
Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?
Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my ...
- 1,776
0
votes
1
answer
241
views
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,...
- 763
0
votes
0
answers
146
views
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 $$(n,d) \mapsto n - \Big(\Big\...
- 48.9k
10
votes
0
answers
416
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
0
votes
1
answer
196
views
Geometric prime distribution
Let integers $\ a>1\ $ and $\ b\in\mathbb Z\ $ be relatively prime (hence $\ b\ne 0).\ $ The Dirichlet's prime distribution theorems apply to the arithmetic sequence
$$ (_aG_b(x) : x\in\mathbb Z) $$...
- 4,786
0
votes
0
answers
177
views
Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?
Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...
- 6,984
1
vote
1
answer
327
views
Symmetry in Hardy-Littlewood k-tuple conjecture
Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...
- 6,984
1
vote
0
answers
77
views
$t$-balanced numbers
Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...
- 6,984
5
votes
0
answers
340
views
On a conjecture about the arithmetic function that counts the number of twin primes
This is cross-posted from the question that I've asked with same title on Mathematics Stack Exchange two months ago, which has remained unanswered.
Given a positive real number $x$ we will write ...
2
votes
0
answers
151
views
Is this conjecture equivalent to Polignac's conjecture?
Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...
- 6,984
4
votes
1
answer
954
views
Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture
The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...
- 1,183
0
votes
1
answer
154
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 ...
- 6,984
-3
votes
1
answer
245
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 ...
- 6,984
1
vote
0
answers
96
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:
$$T_2(n,m)=\sum\limits_{\substack{k=1 \\...
- 1,183
4
votes
0
answers
672
views
Euclides' sieve
This is probably a well-known problem. Given a set or multiset of natural numbers let us construct its "Euclides" closure: on each step we take all possible products $P_i$ of the elements in the set, ...
- 5,063
3
votes
0
answers
138
views
Is the conjunction of Goldbach and NFPR conjecture actually equivalent to Hardy-Littlewood k-tuple conjecture?
In this previous question of mine
I introduce under Goldbach's conjecture the notation $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ as well as the related so-called NFPR conjecture ...
- 6,984
1
vote
1
answer
303
views
Admissible k-tuples and primorials
Let $ (a_{1},\cdots,a_{k}) $ an admissible $ k $ -tuple and $ P_{k} $ the product of the first $ k $ primes. Do we have a conjectural expression for the number of positive integers $ n $ not ...
- 6,984
4
votes
1
answer
531
views
Do prime gaps that are a power of "h" have the same density?
Send me back to Mathematics Stack Exchange if this question is not research level.
At Terence Tao's blog post there is the expression:
$$\sum\limits_{n \leq X} \Lambda(n)\Lambda(n+h) \ \ \ \ \ \ \ \ \ ...
- 1,183
0
votes
2
answers
317
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 ...
user94040
0
votes
1
answer
196
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 ...
- 6,984
4
votes
0
answers
173
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 ...
- 6,984
2
votes
1
answer
269
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 ...
0
votes
3
answers
415
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 ...
- 6,984
1
vote
2
answers
513
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}{\...
- 6,984
2
votes
1
answer
534
views
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 ...
- 2,158