On the Number of Prime Factors of Consecutive Integers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915940538515456 |
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| author | Lau, Cheuk Fung |
| author_facet | Lau, Cheuk Fung |
| contents | We prove that there are infinitely many $n$ such that $ω(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Teräväinen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15042 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Number of Prime Factors of Consecutive Integers Lau, Cheuk Fung Number Theory 11N56 (Primary) 11N36 (Secondary) We prove that there are infinitely many $n$ such that $ω(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Teräväinen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a more efficient sieve procedure with stronger exponential concentration-of-measure estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp. |
| title | On the Number of Prime Factors of Consecutive Integers |
| topic | Number Theory 11N56 (Primary) 11N36 (Secondary) |
| url | https://arxiv.org/abs/2604.15042 |