On the Cramér-Granville Conjecture and finding prime pairs whose difference is 666

Questions

If $p= \text{NextPrime}[q]$ (the smallest prime greater than $p$), and $p-q = 666,$ what are $p$ and $q$?

(There may be multiple choices. I am interested in finding one.)

Cramér-Granville Conjecture:
Defining $p_0=2$, and $p_n$ as the nth odd prime, and the nth prime gap as
$g_n=p_{n+1}-p_n$, then $g_n< M \log(p_n)^2$ for some $M>1$.

Reference link: http://mathworld.wolfram.com/Cramer-GranvilleConjecture.html

If $g_n =666$, what are generally good estimates for $M$ and $p_n$ according to the Cramér-Granville Conjecture? Can we make use of current sieve technology and probabilistic modeling to solve our problem efficiently?

Commentary & Previous Work

If $g_n = 666$ and $p_n = 18691113008663$, then the conjecture is satisfied for any $M>1$.

We let $\text{primegap}_{avg} = x/\pi(x) = g_n = 666$. If $\pi(x) = x/(li(x) + sqrt(x) * log(x)/(8\pi))$, then we have $x/\pi(x) = 666$ which implies $x = 4.73231\times10^{289}$ and $\pi(x) =7.10558\times10^{286}$. So our upper bound estimate of $p_n$ will be less than $4.73231\times10^{289}$.

Therefore, we shall focus our attention on finding consecutive prime pairs whose difference is $666$ in the open interval, $(18691113009329, 4.73231*10^{289})$. And we should expect to find approximately $(c/333)(\pi(4.73231×10^{289})-\pi(18691113009329)) = c *7.10558×10^{286}/333$, or $c * 2.133808*10^{284}$ consecutive prime pairs whose difference is $666$ where $0.5 < c < 1$.

Note: $c \to 1$ as $x\to\infty$ according to the Polignac Conjecture.

Furthermore, we also expect to discover sufficiently many prime gaps greater than $666$ in the open interval, $(18691113009329, 4.73231×10^{289})$, so that the prime gap density or average of $666$ is maintained. And according to the Cramér-Granville Conjecture, the maximum prime gap of $\log(4.73231\times10^{289})^2 = 444892$, more or less, exists in the open interval.

Reference links

https://terrytao.wordpress.com/2009/08/18/the-least-quadratic-nonresidue-and-the-square-root-barrier/#comment-472548;

http://www.ams.org/journals/mcom/1989-52-185/S0025-5718-1989-0947470-1/S0025-5718-1989-0947470-1.pdf/;

https://en.wikipedia.org/wiki/Polignac%27s_conjecture;

https://www.quora.com/What-great-conjectures-in-mathematics-combine-additive-theory-of-numbers-with-the-multiplicative-theory-of-numbers/answer/David-Cole-146;

https://terrytao.files.wordpress.com/2009/08/prime-number-theory1.pdf;

Boeyens, Jan C. A.; Levendis, Demetrius C. (2008), Number Theory and the Periodicity of Matter, Berlin: Springer-Verlag, ISBN 978-1-4020-6659-7;

'A Primer in Density Functional Theory', http://link.springer.com/book/10.1007%2F3-540-37072-2;

https://www.wired.com/2016/11/physicists-uncover-strange-numbers-particle-collisions/.

“Repetition and growth of prime gaps are essential for the efficient generation of the integers.”

Solution 1:

According to the table compiled by Thomas Nicely, $q=18{,}691{,}113{,}008{,}663$ is the first occurrence of $666$ as a prime gap. Nicely attributes the value to a 1989 paper by Jeff Young and Aaron Potler.