**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 $Min[g(n)-f(n)]$.