Expected gaps between prime numbers

TL;DR

This paper investigates the gaps between consecutive prime numbers using elementary recursive methods within Eratosthenes sieve, providing new insights into prime gaps and their distributions, with implications for longstanding open questions.

Contribution

It introduces a recursive relation for prime gaps and constellations, offering a novel elementary approach to estimate prime gaps and address open problems.

Findings

Derived a recursive relation for prime gaps

Estimated occurrence of gaps and constellations between a prime and its square

Provided implications for Erdős and Turán's open questions

Abstract

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erd\"os and Tur\'an.