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 $n>0$?

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