Problem 700

Let

f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).

Characterise those composite $n$ such that $f(n)=n/P(n)$ , where $P(n)$ is the largest prime dividing $n$ . Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$ ? Is it true that, for every composite $n$ ,

f(n) \ll_A \frac{n}{(\log n)^A}

for every $A>0$ ?

Source: erdosproblems.com/700

Number theory
Binomial coefficients

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label. The site credits GPT 5.6 Sol Pro, prompted by Price, with a positive answer to the second question, infinitely many products of three primes with $f(n)\sim n^{2/3}$ (page edited 28 August 2026); see the Price claim page (2026), a pending partial claim. The standing in the frontmatter derives from the claim pages.

Problem 700

Status

Proof claims

Discussion