* Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
* When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question was proved [Bertrand's postulate](https://en.wikipedia.org/wiki/Bertrand%27s_postulate)



A generalization Bertrand's postulate
First of all, as mentioned by Random above, this is a very strong conjecture, because it is stronger than Legendre's conjecture. As far as I know, it is not known even if we assume the truth of Riemann Hypothesis and also some reasonable conjectures on distribution of imaginary parts of zeros, such as the Montgomery's pair correlation conjecture.
However, it is [known](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/83.3.532) that
$$
\pi(x+x^{0.525})-\pi(x)>0
$$
for large $x$. This means, for instance, that if $k$ is large and
$$
n\leq k^{19/21}=k^{0.9047\ldots}
$$
the conjecture is, in fact, true.


Also, there are results on the measure of large gaps. For example,
$$
\sum\_{\substack{p\_n\leq x \\ p\_{n+1}-p\_n>\sqrt{p\_n}}}(p\_{n+1}-p\_n)\ll x^{3/4+o(1)},
$$
see "The Riemann zeta-function: theory and applications" by A. Ivić, Theorem 12.14. This result implies that your conjecture is true for all $n$, except at most $k^{1/2+o(1)}$ for $k\to +\infty$.


