All Questions

We know that $$\liminf_{n\to\infty}{\frac{p_{n+1}-p_n}{\log p_n}}=0.$$ I'm trying to figure out the proof and I have read a lot of documents, I asked a question here. Still I can't see what's going on....

Is the following Open question or Conjecture already known, or eventually settled ? Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...

It is currently an open question if there are infinitely many primes $p$ such that $2p + 1$ is prime (Sophie Germain primes) or that at least one of $24p \pm 1$ is prime. Could Zhang's method, or the ...

I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...