**I know the following:**


* Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
* Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.


**My questions are:**


1. Is the above correct?
2. Has this been conjectured or refuted:


There is a prime number between $n^2$ and $n(n+1)$ for every integer $n>0$ ?
3. What is the minimal range $[f(n),g(n)]$ proven to contain a prime number for every $n>0$?


**UPDATE:**


The term 'minimal' is somewhat confusing in this case, since obviously, the difference between $n$ and $2n$ is in fact smaller than the difference between $n^2$ and $(n+1)^2$.


So to refine my question - 'minimal range' refers to the **ratio** between $f(n)$ and $g(n)$, and not to the difference between them. In other words $Min[g(n)/f(n)]$ instead of $Min[g(n)-f(n)]$.



What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?
Baker-Harman-Pintz (2000) proved that for every sufficiently large $n$ the interval $[n-n^{0.525},n]$ contains a prime number. Schoenfeld (1976) proved that if the Riemann Hypothesis is true, then the term $n^{0.525}$ in the previous statement can be replaced by $\sqrt{n}\log^2 n$. For a very recent strengthening of this result, see this [preprint](http://arxiv.org/abs/1402.6417). This is still probably far from the truth, e.g. Cramér (1937) made the conjecture based on a probabilistic model, that the term $n^{0.525}$ above can be replaced by a certain constant times $\log^2 n$.


