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Claim. There is an absolute constant $C>0$ such that infinitely many positive integers $n$ satisfy $\omega(n+k)\le\Omega(n+k)\le C\log k$ for every integer $k\ge2$. This is Theorem 1.1 of Cheuk Fung (Joshua) Lau, On the number of prime factors of consecutive integers, arXiv:2604.15042 (v1 2026-04-16, v2 2026-06-24), stated for $\Omega$, the number of prime factors counted with multiplicity, and hence for $\omega\le\Omega$. It answers the question of Problem 248 yes after a shift of $n$: for such an $n$ put $n'=n+1$; then for every $j\ge1$,

$$\omega(n'+j)=\omega(n+(j+1))\le C\log(j+1)\le Cj,$$

since $\log(j+1)\le j$ for $j\ge1$, so infinitely many $n'$ have $\omega(n'+j)\ll j$ for all $j\ge1$ with an absolute implied constant. The paper's random models (Conjectures 5 and 6) say that the bound $C\log k$ is sharp up to the constant. The proof refines the probabilistic sieve argument of Tao and Teräväinen: $n$ is drawn from $[x,2x]$ weighted by a product of Selberg sieve weights with polynomially decaying levels $R_k=x^{c/k^{50}}$, and a sieve-weighted concentration estimate for $\Omega(n+k)$ (Proposition 5.5) makes the union bound over $k\ge2$ converge. Theorem 1.3 gives the same bound for $n-k$. The source card is Lau 2026.

Depends on. Nothing in this wiki; the argument is self-contained in the preprint. It refines the method of Tao and Teräväinen 2025 but does not rest on that result.

Acceptance. Reviewed: the site's curator, Thomas F. Bloom, labels the problem PROVED (LEAN) and credits Lau [La26] in the problem's commentary with improving the bound to $\omega(n+k)\ll\log k$ for all $k\ge2$, noting that $\omega$ may be replaced by $\Omega$ (page last edited 2026-04-17, after a thread comment of the same day cited the preprint, which the site marks as addressed); that documented acceptance is the reviewed evidence. Not refereed: the preprint has no journal publication on record. Not formalized: no Lean development of this result is known; the lean-proofs file linked from the Tao–Teräväinen page formalizes their linear bound, not this one.