Gaps of powers of consecutive primes and some consequences

TL;DR

This paper investigates inequalities involving powers of consecutive primes, establishing their infinite occurrence through an extension of Kummer's characterization of series convergence.

Contribution

It introduces a new inequality involving powers of consecutive primes and proves its infinite validity using an extended form of Kummer's criterion.

Findings

The inequality holds for infinitely many n.

Extension of Kummer's characterization is key to the proof.

Provides insights into the distribution of prime gaps.

Abstract

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.