Linked Questions
11 questions linked to/from Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?
111
votes
30
answers
65k
views
Open problems with monetary rewards
Since the old days, many mathematicians have been attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express their ...
79
votes
6
answers
11k
views
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...
- 79.9k
9
votes
3
answers
584
views
Why is there an unexpected increase in the density of certain types of Goldbach primes?
Note: Posted in MO since it was unanswered in MSE.
I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the ...
- 5,460
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 ...
- 6,976
6
votes
0
answers
726
views
Would the following conjectures imply Cramer's conjecture?
Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
- 6,976
-1
votes
1
answer
162
views
Upper bound for the number of $k$-central numbers in a prime gap
Let $I_{n}:=]p_{n},p_{n+1}[$ be the open interval between the $n$-th and $(n+1)$-th prime. Under Goldbach's conjecture, denote by $r_{0}(m)$ the smallest positive integer $r$ such that both $m-r$ and $...
- 6,976
4
votes
0
answers
143
views
Can this number be interpreted as a fractal dimension?
Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$.
Let's ...
- 6,976
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 ...
- 6,976
3
votes
0
answers
134
views
Distribution of the inbetween prime
Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and
$$2=p_0<p_1<\ldots$$
be the increasing sequence of all primes. Let
$$ \forall_{n=1\ 2\ \ldots}\ \ d_n\...
- 7,500
2
votes
0
answers
84
views
The number of admissible tuples with last element equal to $h_{k-1}$?
Let $k \geq 2$ and $(h_1, h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$.
Consider the $k$-tuple : $\mathcal{H}_k=(0,h_1,\cdots,h_{k-1})$ with $0<h_1<\cdots<h_{k-1}$.
The $k$-tuple $\mathcal{H}...
- 521