Problem 685

../

---

Statement. Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is

$$(1+o(1))k\sum_{k<p<n}\frac{1}{p}?$$

Or perhaps even when $k \geq (\log n)^c$?

Status. Open.

Source. erdosproblems.com/685, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #685, https://www.erdosproblems.com/685.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.