Large gaps between consecutive prime numbers containing perfect powers

Abstract

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$c_k\frac{\log p{\log }_{2}p{\log }_{4}p}{({\log }_{3}p{)}^2},$$ where $p$ is the smaller of the two primes.