Chains of large gaps between primes

TL;DR

This paper proves that for any fixed number of consecutive large prime gaps, there are infinitely many chains of such gaps with size growing roughly like a logarithmic factor, extending previous results for single gaps.

Contribution

It extends prior work on large prime gaps by establishing the existence of arbitrarily long chains of large gaps with effective bounds, using a combination of recent methods and the Maier matrix technique.

Findings

For fixed k, chains of k large prime gaps occur infinitely often.

The size of these chains grows proportionally to (log X * log log X * log log log log X) / log log log X.

The bounds are effective and independent of k.

Abstract

Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$ G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that \[ G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that \[ G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.