I am searching for an lower bound on the difference between the $(n+k)$-th and $n$-th prime number in terms of $k$. I have something like this in mind (conjecture):
Let $(p_k)_k$ denote the sequence of primes (i.e. $p_1 = 2, p_2 = 3, p_3 = 5, \dotsc$). Then there exists a constant $c > 0$, such that $p_{n+k} - p_n \ge c \cdot p_k$ for all $n,k \in \mathbb{N}$.
or alternatively
Let $(p_k)_k$ denote the sequence of primes (i.e. $p_1 = 2, p_2 = 3, p_3 = 5, \dotsc$). Then there exists a constant $c > 0$, such that $p_{n+k} - p_n \ge c\cdot ( k \cdot \log k)$ for all $n,k \in \mathbb{N}$.
Having the prime number theorem, more precisely $p_n \sim n \log n$, in mind, this seems to be true... I appreciate any hint.