On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914578815778816 |
|---|---|
| author | Campbell, Peter J. |
| author_facet | Campbell, Peter J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10356 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares Campbell, Peter J. Number Theory 11N05, 11N36, 11Y11, 11Y16 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. |
| title | On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares |
| topic | Number Theory 11N05, 11N36, 11Y11, 11Y16 |
| url | https://arxiv.org/abs/2603.10356 |