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

Abstract

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n\ge 1$, 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 $n^2\le 10^{31}$, 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.

15 pages, no figures. Revised version with expository changes