Large gaps between consecutive prime numbers containing perfect powers

TL;DR

This paper proves that infinitely often, large prime gaps contain perfect powers, with the size of these gaps growing proportionally to a specific logarithmic function of the prime.

Contribution

It establishes the existence of infinitely many prime gaps containing perfect powers, with explicit bounds on the gap size related to logarithmic functions.

Findings

Infinitely many prime gaps contain perfect powers.

Gaps of size proportional to rac{\u2113 p rac{2}{p} rac{4}{p}}{(rac{3}{p})^2} occur infinitely often.

Perfect powers appear inside very long prime gaps.

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.