Erdős problem #396

For each integer $k\ge 1$ , determine the smallest integer $n\ge k$ such that $n(n-1)(n-2)\cdots(n-k)$ divides the central binomial coefficient $\binom{2n}{n}$ ; equivalently, determine $n_k=\min\{n\in\mathbb{Z}_{\ge k}:\prod_{j=0}^{k}(n-j)\mid\binom{2n}{n}\}$ whenever this set is nonempty.

Progress summary

Partially solved

A new computation records the smallest known examples in a finite range, but it does not settle the general Erdős problem.

Erdős problem #396 is a named Erdős question whose general resolution remains open. The latest development is a computational record of smallest witnesses for $k=8,\ldots,13$ .