Erdős Problem #679

The problem

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?

Statement quoted from erdosproblems.com/679, maintained by Thomas Bloom. Original source: [Er79d]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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