# Least prime $p_n$ such that $p_{n+1}-p_n\geq k$

I define $\rho(k)$ to be the least prime $p_n$ such that $p_{n+1}-p_n\geq k$.

Since every number in the set $\{(k+1)!+2, (k+1)!+3, (k+1)!+4, \cdots, (k+1)!+(k+1)\}$ is composite, we have that $\rho(k)\leq (k+1)!+1$.

Are there better bounds for it?

Thanks.