On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares

TL;DR

This paper proves that every interval between consecutive squares contains an integer with at most three prime factors, improving previous results by combining computational methods and explicit sieve-theoretic arguments.

Contribution

It provides an explicit proof of an analogue of Legendre's conjecture for almost primes, reducing the prime factor count from four to three in these intervals.

Findings

Intervals between consecutive squares contain integers with ≤3 prime factors.

The result holds for all n ≥ 1, with explicit bounds for small and large n.

The proof combines computational data and explicit sieve methods.

Abstract

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the previous best result of Dudek and Johnston, who showed that every such interval contains an integer with at most $4$ prime factors. The proof is divided into two ranges. For $n^2 \leq 10^{31}$, we use prior computational results on primes in short intervals between consecutive squares, together with explicit bounds on maximal prime gaps. For $n^2 > 10^{31}$, we give a sieve-theoretic argument with explicit constants, adapting Richert's logarithmic weights to intervals between consecutive squares and employing an explicit linear sieve of Bordignon, Johnston, and Starichkova.