$p_i$ is the $i^{\rm th}$ prime. $\pi(x)$ is prime counting function.
Firstly, I think that Prime gap inequality holds true for any $i>0$: $p_{i+1} - p_{i} \leq i$.
Very often, $\pi(p_{m}+m) - m \leq \pi(p_{n}+n) - n$ if $m
Conjecture:$ \forall x , \exists m,n$, satisfy $ x