Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers

TL;DR

This paper proves that there are infinitely many pairs of consecutive primes with large gaps, each containing a perfect $k$-th power of a prime, revealing new insights into prime distribution and prime powers.

Contribution

It establishes the existence of infinitely many large prime gaps that include prime powers, a novel result linking prime gaps and prime powers.

Findings

Infinitely many prime pairs with gaps exceeding a specific logarithmic bound.

Each such interval contains a prime power of a fixed order $k$.

Demonstrates the interplay between large prime gaps and prime powers within those gaps.

Abstract

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.