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Claim. Theorem 7.3 of Cheuk Fung Lau, On the number of prime factors of consecutive integers, arXiv:2604.15042, carded at Lau 2026, in Section 7, titled by the paper as the conditional falsity of Erdős Problem 679. It assumes the paper's Conjecture 8: for some $C_0\ge1$ and some $d$ with $1\le d<C_0$, every interval $(x-(\log(x/2))^d,x]$ with $x$ large contains an integer $m$ with $\omega(m)\ge C_0\log\log m/\log\log\log m$. Under that assumption there is $\delta>0$ such that every sufficiently large $n$ has some large $k<n$ with

$$\omega(n-k)>(1+\delta)\frac{\log k}{\log\log k}.$$

So, under the hypothesis only, the first question of Problem 679 has the answer no for every $\epsilon<\delta$: only finitely many $n$ have $\omega(n-k)<(1+\epsilon)\log k/\log\log k$ for all large $k<n$. The proof applies the conjecture at $x=n$ and compares $C_0\log\log(n-k)/\log\log\log (n-k)$ with $\log k/\log\log k$ through $k\le(\log(n/2))^d$, which gives the bound with any $\delta<C_0/d-1$. The claim is conditional: Conjecture 8 is unproven, so this page derives nothing for the problem's standing. Section 7 is unchanged in the paper's second version (2026-06-24). The paper's unconditional Theorem 1.3, that for some $C$ infinitely many $n$ have $\omega(n-k)\le\Omega(n-k)\le C\log k$ for all $1<k<n$, settles no instance of the problem and is progress, not a claim; its Conjecture 6, that the $C\log k$ bound is sharp up to a constant, which would also give a negative answer, is only a conjecture.

Depends on. Nothing in this wiki.

Standing. Claimed, conditional. No refereed version was found; the site's remarks (page last edited 17 April 2026) record Theorem 1.3 and the conjecture, and the site labels the problem OPEN.