  Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely
many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and
$$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n},
$$ for some small constant $c_m>0$. Furthermore, we also obtain a related
result concerning the least primes in arithmetic progressions.
