On a conjecture of Erd\H{O}s, P\'olya and Tur\'an on consecutive gaps between primes

TL;DR

This paper proves a longstanding conjecture by Erdős, Pólya, and Turán that certain linear combinations of consecutive prime gaps change sign infinitely often, using advanced modern methods in number theory.

Contribution

The paper confirms the conjecture on sign changes of linear combinations of prime gaps, employing the Maynard-Tao method and recent developments in prime gap research.

Findings

Confirmed the conjecture on sign changes of linear combinations of prime gaps.

Applied Maynard-Tao method to analyze prime gap linear combinations.

Extended recent work of Banks, Freiberg, and Maynard to this problem.

Abstract

Let p_n denote the sequence of all primes and let d_n=p_n-p_{n-1} denote the sequence of all gaps between consecutive primes. In 1948 Erd\H{o}s and Tur\'an showed that d_{n+1}-d_n changes sign infinitely often and together with P\'olya asked for a necessary and sufficient condition that a fixed linear combination of r>1 consecutive prime gaps d_{n+i} i=(1,2,...r) should change sign infinitely often as n runs through the sequence of all natural numbers. They conjectured that the condition is that the non zero values among the coefficients of the linear combination cannot all have the same sign but they could prove only much weaker relations. In the present work this conjecture is proved by the aid of the Maynard-Tao method and other important ideas of the recent work of W.D. Banks, T. Freiberg and J.Maynard (see arXiv:1404.59094v2).