Conjecture:
Given $\varepsilon>0$ there are only a finite number of primes $p_n$ such that $p_{n+1}-p_n>p_n^{\,\varepsilon}$.
Do anyone have an idea if this can be proved or not?
The method used is to test all primes $< 100,000,000$ and calculate the number of hits and the largest prime in the set of hits.
$\quad\varepsilon\quad\quad\quad\quad p_{max}\quad\quad hits$
0.50 113 6
0.40 31,397 41
0.30 47,326,693 2,003
0.25 99,988,649 54,726
For $\;\varepsilon=0.25\;$ is so close to the limit ($100,000,000$) that $p_{max}$ and $hits$ probably are much greater than in the diagram.