Gaps of powers of consecutive primes and some consequences

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.

7 pages