Linked Questions

109 votes
29 answers
62k 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 ...
Noam D. Elkies's user avatar
8 votes
3 answers
563 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 ...
Nilotpal Kanti Sinha's user avatar
0 votes
3 answers
413 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 ...
Sylvain JULIEN's user avatar
6 votes
0 answers
724 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+...
Sylvain JULIEN's user avatar
0 votes
1 answer
144 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 $...
Sylvain JULIEN's user avatar
4 votes
0 answers
141 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 ...
Sylvain JULIEN's user avatar
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 ...
Sylvain JULIEN's user avatar
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\...
Włodzimierz Holsztyński's user avatar
1 vote
0 answers
103 views

Tiling the set of integers with intervals of the form $[n-r_{0}(n),n+r_{0}(n)]$

Assuming Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ as well as $p_{\pm}(n):=n\pm r_{0}(n)$. Consider a sequence $(c_{m})_{m>0}$ defined by $c_{1}:=4$ and $c_{...
Sylvain JULIEN's user avatar
2 votes
0 answers
83 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}...
Lagrida Yassine's user avatar