Problem 679

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claims/: The 2 claim pages of Problem 679, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that

$$\omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k}$$

for all $k<n$ which are sufficiently large depending on $\epsilon$ only?

Can one show the stronger version with

$$\omega(n-k) < \frac{\log k}{\log\log k}+O(1)$$

is false?

Status. Open, the site's label (page last edited 17 April 2026). The site's remarks credit the forum user DottedCalculator with disproving the stronger version, the second question, and record Lau's unconditional bound; the standing in the frontmatter derives from the claim pages under the parts epsilon_version and stronger_version: the second question has the pending partial claim on DottedCalculator's page, and the first has only the conditional result on Lau's page, so the problem stays open.

Source. erdosproblems.com/679, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #679, https://www.erdosproblems.com/679.

References.

• [La26] C. F. Lau, On the number of prime factors of consecutive integers. arXiv:2604.15042 (2026).

Formalization. No formal-conjectures statement is recorded for this problem. A Lean proof of the disproof of the stronger version, assuming the asymptotic $p_n=(1+o(1))\,n\log n$, is linked on DottedCalculator's claim page; this corpus has not built it, so it gives no formalized evidence.

Current assessment

The questions (site formulation accessed 2026-09-04; page last edited 17 April 2026). The statement above, two questions about the integers $n$ whose predecessors $n-k$ all have few distinct prime factors. The first asks whether, for each $\epsilon>0$, infinitely many $n$ have $\omega(n-k)<(1+\epsilon)\log k/\log\log k$ for all $k<n$ large in terms of $\epsilon$; the second asks whether the version with $\log k/\log\log k+O(1)$ in place of the factor $1+\epsilon$ is false. The site's label is OPEN. The remarks add that the analogous questions can be asked for $\Omega$, with $\log k/\log 2$ in place of $\log k/\log\log k$.

The second question. Answered yes, that is, the stronger version is false, by the primorial argument on DottedCalculator's claim page: for every constant $C$, every large $n$ has some $k<n$ with $\omega(n-k)\ge\log k/\log\log k+C$, and the site's remarks state the sharper form with $c\log k/(\log\log k)^2$ in place of $C$. The site credits the result in its remarks but labels the problem OPEN, so the claim stays claimed; a Lean proof of it, assuming the prime number theorem's asymptotic for the $n$th prime, is linked on that page and is not built here.

The first question. Open. Lau [La26], Theorem 1.3, proves that for some constant $C$ infinitely many $n$ have $\omega(n-k)\le\Omega(n-k)\le C\log k$ for all $1<k<n$, within a factor $\log\log k$ of the bound asked for, and conjectures that the $C\log k$ bound is sharp up to a constant, which would give the answer no; his Theorem 7.3 proves the answer no for every $\epsilon$ below some $\delta>0$ under a conjecture on short intervals containing integers with many prime factors, the conditional claim on Lau's claim page. Neither result settles an instance of the question.

Search scope. The site's problem page, its discussion thread (the posts of 11 and 12 January 2026) and the two arXiv versions of [La26]; the site's proof-claims tab lists no claim for the problem; no literature database searched.