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

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.