Almost primes between all squares

TL;DR

This paper proves that between every pair of consecutive squares, there exists a number with at most four prime factors, using explicit sieve methods, extending previous results to all natural numbers.

Contribution

It provides the first universal proof for all n ≥ 1, employing explicit linear and Kuhn's weighted sieve techniques for generic sifting sets.

Findings

Existence of a number with ≤4 prime factors between consecutive squares for all n≥1

Explicit version of Kuhn's weighted sieve developed for generic sifting sets

Utilizes recent computations by Sorenson and Webster

Abstract

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods.