TITLE: Relative sizes of prime gaps
QUESTION [8 upvotes]: There are no prime numbers between the two primes $113$ and $127$.  That gap seems quite large by comparison to the sizes of the numbers in it.
$$
\frac{\text{size of gap}}{\text{prime just below the gap}} = \frac{14}{113} = 0.12389\ldots
$$
Is that the biggest that this particular statistic ever gets?
Is that the last time this particular statistic ever gets that big?

REPLY [2 votes]: Rohrbach and Weis show that (as Martin notes in the comments):
$$ g_n < \frac{p_n}{13} \quad \quad n > 118$$
From this we have that your statistic is bounded by: $ \frac{1}{13} \approx 0.0769231 $ for $n > 118$. The 118th prime is $647$, and a quick numerical analysis verifies this.
Mathematica code to verify this (which should print nothing):
Do[If[Prime[k + 1]/Prime[k] - 1 >= 14/113, Print[k]], {k, 31, 119}]