Problem 679

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?

Source: erdosproblems.com/679

Number theory

Wiki page

Status

Open

No claim settles this problem.

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.

Problem 679

Status

Proof claims

Discussion