Bounded gaps and perfect power gaps in sequences of consecutive primes

TL;DR

This paper investigates the simultaneous behavior of consecutive prime gaps, exploring whether they can all be large, all perfect powers, or none, and presents related results and open problems.

Contribution

It introduces new questions and partial results on the structure of consecutive prime gaps and their relation to perfect powers.

Findings

Certain configurations of prime gaps and perfect powers are shown to be impossible.

New bounds and conditions for prime gaps related to perfect powers are established.

Open problems and conjectures are proposed for future research.

Abstract

We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also presented.