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

Abstract: We prove an explicit almost-prime analogue of Legendre's conjecture. 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 combines a finite verification for $n^2 \leq 10^{31}$, obtained from computations on primes in short intervals between consecutive squares together with explicit bounds on maximal prime gaps, with a fully explicit sieve-theoretic argument for the remaining range. For large $n$, we adapt Richert's logarithmic weights to intervals between consecutive squares and employ an explicit linear sieve of Bordignon, Johnston, and Starichkova.