Suppose $p_n$ is $n$-th prime, $g_n=p_{n+1}-p_n$ is the corresponding prime gap. What is the highest number $C$, such that $p_N>C$ can be proven for $N=\min\{n\mid g_n\geq 1.609\cdot 10^{18}\}$.

**Motivation:** I've read about Goldbach's weak conjecture. The number $C$ above is obvious lower bound for the first odd number, which does not admit a representation as a sum of three primes, which follows from check of Goldbach's conjecture up to $1.609\cdot 10^{18}$, which is done already by computers. I just want to know, how big is it.