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arXiv · 1709.02283

Gaps of powers of consecutive primes and some consequences

Douglas Azevedo·Tiago Reis
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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.

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Douglas Azevedo, Tiago Reis. 2017-09-07. Gaps of powers of consecutive primes and some consequences. https://arxiv.org/abs/1709.02283

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