The largest possible prime gap?

What is the largest possible prime gap if we observe only 1000-digits numbers? There are few conjectures about this question but is there something that we can say and be absolutely sure of it?

Solution 1:

If $a=A(n)$ is the $n^{th}$ primorial (that is, the product of the first n primes), then at least $a,a\pm2, a\pm3,... a\pm p_n,a\pm (p_n+1)$ are composite, giving a prime gap of length at least $p_n-1$. At $n=350, p_n=2357$, the primorial $a=A(n)$ has 1000 digits (its base 10 logarithm is about 999.375). Actually, with Pari/GP it's easy to verify that $a-4152 ... a+3312$ are composite and $a-4153$ and $a+3313$ apparently are prime, giving a prime gap of length 7465.

Wikipedia gives a result of R. Rankin, that (where $g(p)$ denotes the gap after $p$) $g(p) > 2e^\gamma (\ln p) (\ln \ln p) (\ln \ln \ln \ln p)/(\ln \ln \ln p)^3$ infinitely often. For $p\approx 10^{1000}$, that expression evaluates to about 5300. The article also gives a conjectured bound of H. Cramér, that $g(p) = O((\ln p)^2)$, which for 1000-digit numbers would be some multiple of 5 million.

For a broader discussion with numerous references, see the Prime Gaps page at wolfram.com. For more-recent results on lower bounds for maximal prime gaps, see the arXiv.org paper or preprint Long gaps between primes by Ford, Green, Konyagin, Maynard, and Tao, and see Tao's commentary at terrytao.wordpress, and see additional results / background at the same site.