TITLE: What is the best known estimate for the place of the prime gap with length 1.609*10^18?
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: Well, Wikipedia's page on Bertrand's postulate which Scott referred to in his answer  does not cite the strongest estimates on this problem. The paper of Olivier Ramaré and Yannick Saouter MR 2004a:11095 "Short effective intervals containing primes. 
J. Number Theory 98 (2003), no. 1, 10–33 yield stronger result. They prove that
the interval $[x(1-1/\Delta), x]$ contains a prime for $x \geq 10 726 905 041$ and $\Delta=28 313 999$. In fact they have a table that gives even stronger result in the relevant range for this problem. If we look at  Table 1 from their paper we get that for $x \geq e^{60}$ we can choose $\Delta=209 267 308$. We thus need to determine $x$ so that
$$ \frac x {209 267 308} <1.609 \cdot 10^{18}$$
Computation shows that $x<3.367\cdot 10^{26}$. Since $\log(3.367\cdot 10^{26})=61.1>60$ this is permissable. This gives a better estimate than Charles and Scott's answers above.
It should be remarked that already Ramaré-Saouter used these estimates for the Ternary Goldbach problem. However due to a shorter range where the Goldbach problem had been checked at that time they did get the  shorter range $1.13256\cdot 10^{22}$ (Corrollary 1) than using the recent computer checked bounds for the Goldbach problem.