Problem 699

Erdős Problem #699 [status: falsifiable; formalized: yes]. Is it true that for every $1\leq i<j\leq n/2$ there exists some prime $p\geq i$ such that$$p\mid \textrm{gcd}\left(\binom{n}{i}, \binom{n}{j}\right)?$$ Current best: A theorem of Sylvester and Schur says that for any $1\leq i\leq n/2$ there exists some prime $p>i$ which divides $\binom{n}{i}$. Erd\H{o}s and Szekeres further conjectured that $p\geq i$ can be improved to $p>i$ except in a few special cases. They also found some counterexamples when $i=3$, but only one counterexample when $i\geq 4$:$$\textrm{gcd}\left(\binom{28}{5},\binom{28}{14}\right)=2^3\cdot 3^3\cdot 5.$$This is mentioned in problem B31 of Guy's collection [Gu04]. Prize: no. Tags: binomial coefficients, number theory.