Gap primes are certainly even numbers. (starting from the consecutive primes $3$ and $5$)
Let $n$ be however an odd number and define by $n^2$ the range of the examined primes ( i.e. the range is defined from $2$ to $n^2$)
Now I have examined gaps up to the value of $71^2=5041$.
It seems that in all these ranges defined by $n^2$ there is only one special gap equal to $14$ which occurs in the special range up to $13^2$.
It is special because $14$ is greater than $13$.
The gap is between numbers $113$ and $127$.
For others ranges defined by $n^2$ the maximal gap is less than $n$.
My question:
Is the gap between $113$ and $127$ unique for set of all primes, or however we can find the next such special gap with property described above?
If not how to explain it?
Using https://oeis.org/A000101 and https://oeis.org/A005250 , I did check that this seems the only such special gap. And looking at the trend in
, it doesn't look like there will be any other later too. This doesn't seem provable with the current results on prime gaps though (https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds). We require $g_n < p_n^{0.5}$ to prove this, but the best we currently have is $g_n < p_n^{0.525}$