Problem 698

Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$

\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?

Source: erdosproblems.com/698

Number theory
Binomial coefficients

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

The site labels the problem PROVED (LEAN). The standing derived from the claim page is solved, proved, by Bergman 2011, whose bound tends to infinity with $n$ uniformly in $i$ (the claim page derives the explicit $h(n)=n^{1/2}/6$ from it); a refereed paper credited by the site's curator. The Lean proofs the site's label refers to are third-party work not built here.

Problem 698

Status

Proof claims

Discussion