Increasing and decreasing prime gaps

TL;DR

The paper proves the existence of infinitely many long decreasing and increasing sequences of prime gaps, settling a conjecture of Erdős for sequences of length two, and explores the behavior of prime gaps.

Contribution

It establishes the existence of arbitrarily long monotonic sequences of prime gaps, including the case of length two, confirming Erdős's conjecture.

Findings

Existence of infinitely many sequences of decreasing prime gaps

Existence of infinitely many sequences of increasing prime gaps

Confirmation of Erdős's conjecture for sequences of length two

Abstract

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed inequalities. In doing so we settle a conjecture of Erd\"os for the case $m=2$.