Work

Almost primes between all squares

Year: 2025

Type: article

Abstract: We prove that for all n ≥ 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 ≥ 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.

Source: Journal of Number Theory

DOI: 10.1016/j.jnt.2025.05.009

Authors: Adrian W. Dudek, Daniel R. Johnston

Institutions: The University of Queensland, UNSW Sydney, UNSW Canberra

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: Australian Government