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

Peter Campbell

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.

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 , the interval contains an integer having at most 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 prime factors. The proof combines a finite verification for , 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 , we adapt Richert's logarithmic weights to intervals between consecutive squares and employ an explicit linear sieve of Bordignon, Johnston, and Starichkova.
Paper Structure (6 sections, 14 theorems, 108 equations)

This paper contains 6 sections, 14 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of sieving
  3. An explicit linear sieve
  4. Richert's weighted sieve
  5. An upper bound for the weighted sifting term
  6. Proof of Theorem \ref{['thm:main_omega3']}

Key Result

Theorem 1.1

For every integer $n\geq 1$, there exists $a\in(n^2,(n+1)^2)$ such that $\Omega(a)\leq 3$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2: Explicit version of the linear sieve bordignon_johnston_starichkova2025
  • Lemma 2.3
  • Lemma 2.4: rosser_schoenfeld1962
  • Lemma 2.5
  • proof
  • Lemma 3.1: johnston2026primesprimescubes
  • Lemma 3.2: johnston2026primesprimescubes
  • ...and 9 more