All Questions
12 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
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
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
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 ...
0
votes
1
answer
148
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 ...
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
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 ...
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