Goldbach versus de Polignac numbers

TL;DR

This paper explores the relationship between Goldbach and de Polignac numbers using sieve theory, showing that under certain assumptions, either Goldbach numbers are closely spaced or de Polignac numbers are dense, and also analyzes prime gap limit points.

Contribution

It introduces a conditional framework linking Goldbach and de Polignac numbers and provides a new density result for prime gap limit points based on recent sieve theory advances.

Findings

Either Goldbach numbers are finitely close or de Polignac numbers have full density.

The set of limit points of normalized prime gaps has density at least 2/3.

Conditional results depend on primes having a level of distribution greater than 1/2.

Abstract

In this note we use recent developments in sieve theory to highlight the interplay between Goldbach and de Polignac numbers. Assuming that the primes have level of distribution greater than $1/2$, we show that at least one of two nice properties holds. Either consecutive Goldbach numbers lie within a finite distance from one another or else the set of de Polignac numbers has full density in $2 \mathbb{N}$. Using very similar techniques we give a conditional proof that the set of limit points of the sequence of normalised prime gaps $(p_{n+1}-p_n)/ \log p_n$ has density at least $2/3$ in the positive reals.