Suppose $p\_n$ is $n$-th prime, $g\_n=p\_{n+1}-p\_n$ is the corresponding [prime gap](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Goldbach%27s_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. 



What is the best known estimate for the place of the prime gap with length 1.609*10^18?
The title and body ask different questions, so I'll address both.


A reasonable estimate for the first prime gap of length $L$ is $e^{\sqrt L}$, so no prime gap this large would be expected below $\exp(1.268\cdot10^9)$, a 550,886,759-digit number.


As for a lower bound, Dusart 2010 [1] shows that for $x\ge396,738$, there is a prime between x and $x\left(1+\frac{1}{25\log^2x}\right)$, so 113353896002617492536754 (about $1.13\cdot10^{23}$) is a lower bound.


Under the Riemann hypothesis ([2]), the bound can be improved to 15373988432858515871940264945439 (about $1.5\cdot10^{31}$).


[1] <http://arxiv.org/abs/1002.0442>


[2] Lowell Schoenfeld, 'Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$. II'. Mathematics of Computation, Vol 30, No 134 (Apr 1976), pp. 337-360.


