What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$? 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:


*

*Is the above correct?

*Has this been conjectured or refuted:
There is a prime number between $n^2$ and $n(n+1)$ for every integer $n>0$ ?

*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 $Min[g(n)-f(n)]$.
 A: 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. 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 $\log^2 n$.
A: Yes what you say is correct. 
For the minimal range known, it is know by results of Baker, Harman, Pintz that $[x, x + x^{0.525}]$ contains a prime for all sufficiently large $x$, as a consequence of this, but in fact already of earlier results, it is known that there is a prime between $n^3$ and $(n+1)^3$ for all sufficiently large $n$. If one could replace $0.525$ by $0.5$ this would essentially yield the conjecture you mentioned, known as Legendre's conjecture.   
This also immediately yields, but really for this the Prime Number Theorem suffices, that the ratio of your $g(n)/f(n)$ can be taken to tend to $1$ as $n \to \infty$. And, the difference, in a suitable sense, is rather the better notion to consider here. 
It is conjetured that $[x, x + f(x)]$ contains a prime for each $x$ where $f$ is some  $O( (\log x)^2)$; the key-word here is Cramér's conjecture, which is still slightly more precise but it is not quite clear what exactly should be conjectured. But the $O( (\log x)^2)$ is likely about alright.
In the converse direction it is known (due to Rankin) that there are infinitely many $n$ such that the gap between $n$ and $n+1$ prime is of size 
$$c \frac{(\log n) (\log \log n)(\log \log \log \log n)}{ (\log \log \log n)^2} $$
for a positive $c$. 
