Problem 396

Erdős Problem #396: declared status 'open'. Formalized: yes. Is it true that for every $k$ there exists $n$ such that$$\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$$ Current best: Pomerance [Po14] has shown that for any $k\geq 0$ there are infinitely many $n$ such that $n-k\mid\binom{2n}{n}$, although the set of such $n$ has upper density $<1/3$. Pomerance also shows that the set of $n$ such that$$\prod_{1\leq i\leq k}(n+i)\mid \binom{2n}{n}$$has density $1$. Prize: no. OEIS: A375077. Tags: binomial coefficients, number theory.