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.

References

Primary source

GitHub

Additional references

Erdős Problem 396 computational project — GitHub

combinatorics computational-mathematics

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

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 $k=8,\ldots,13$ .

August 2026 finite-range computation

The project supplies data for future general arguments, but explicitly does not resolve the problem beyond the tested finite range.

Current status (as of August 2026): Finite-range computational data are available for $k=8,\ldots,13$ , while the general Erdős problem remains open.

Sources

github.com

No solutions have been posted yet.