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.
Best Answer
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.