On the Number of Prime Factors of Consecutive Integers

TL;DR

The paper proves the existence of infinitely many integers with a bounded number of prime factors relative to their shifts, improving previous bounds and advancing questions posed by Erdős.

Contribution

It introduces a refined probabilistic and sieve-based method to bound the number of prime factors of shifted integers, improving on Tao-Teräväinen's results.

Findings

Proved infinitely many n with ω(n+k) ≪ log k for all k ≥ 2.

Enhanced probabilistic sieve techniques with stronger concentration estimates.

Formulated a conjecture on integers with many prime factors based on Cramér-type models.

Abstract

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erd\H{o}s. The proof is based on a quantitative refinement of the Tao-Ter\"{a}v\"{a}inen 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 Cram\'{e}r-type random models. Assuming this conjecture, the main bound is essentially sharp.