Say that the $n$-th prime $p\_n$ is *isolated to degree $k$* 
(my notation) if
the prime gap to either side is larger than $\log p\_n$ to the $k$-th power:
\begin{eqnarray\*}
p\_n - p\_{n-1} & > & (\log p\_n)^k \;,\\
p\_{n+1} - p\_n & > & (\log p\_n)^k \;,
\end{eqnarray\*}
where $\log$ is the natural log.


*Examples*. 


* For $k=1.5$, 
$p\_{4059}=38501$ is isolated because
$$(p\_{n-1},p\_n,p\_{n+1}) = (38461,38501,38543)\;,$$
and the gaps of $40$ and $42$ both exceed 
$(\log 38501)^{3/2} \approx 10.6^{1.5} \approx 34.3$.
* For $k=1.6$,
$p\_{722697}=10938023$ is isolated because
$$(p\_{n-1},p\_n,p\_{n+1}) = (10937921,10938023,10938119)\;,$$
and the gaps of $102$ and $96$ both exceed $(\log 10938023)^{1.6} \approx 86.2$.
* For $k=1.7$, I find no isolated primes in the first
$10$-million primes.
(The $10$-th million prime is $179424673$.)
* Among the first
$10$-million primes, about $13$% are isolated to degree $k = 1$,
and $73$% are isolated to degree $k=\frac{1}{2}$.



> 
> ***Q***. For which $k$ are there an infinite number of isolated primes of degree $k$?
> 
> 
> 



Primes isolated by large gaps to either side
Currently it is not even known, for any $k>1$, if $p\_{n+1}-p\_n>(\log p\_n)^k$ holds infinitely often. The best known result in this direction is due to Ford, Green, Konyagin, Maynard, Tao, see [here](http://arxiv.org/abs/1412.5029).


On the other hand, as Jeremy Rouse explained in two comments, it is expected that the original inequalities hold infinitely often for any $k<2$.


