Is the largest gap between consecutive primes less than the first $27,000$ integers equal to $52?$

Question

Is the largest gap between consecutive primes below the first $27,000$ integers equal to $52?$

At what point does a gap greater than $52$ occur?

I tried analyzing a formula due to Maynard, Tao, and Ford to find the answers:

$$ \frac{\log X \log \log X \log \log \log \log X}{\log \log \log X} $$

I'm getting a negative number from this formula so I'm having trouble finding the answer.

Answer 1

The function:

$$\Psi_g(X)= e^{\frac{g}{\log(X)}}=X $$

provides a decent measure for the largest gap $g$ between consecutive primes less than $X.$

It is not the best, because it is off by $52$ when $X=27,000$ and off by $417$ when $X=2^{64}.$

Answer 2

The first gap of $52$ is:

$$p_{2226}-p_{2225}=19661-19609=52$$

The first gap larger than $52$ is:

$$p_{3386}-p_{3385}=31469-31397=72$$