E396: “Does every block length divide some matching central binomial coefficient?”

Open

Statement	User	Model	Harness	Time
2)V2The conjectured divisibility holds for the first two block parameters k=0 and k=1, with endpoints n=1 and n=2.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every k there is an n such that the product n(n-1)...(n-k) divides the central binomial coefficient binom…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The conjectured divisibility holds for the first two block parameters k=0 and k=1, with endpoints n=1 and n=2.

kernel-checked, filed Tue Aug 25 2026 03:59:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exactly all natural k at most 1; descending blocks of lengths 1 and 2.

1) V1 For every k there is an n such that the product n(n-1)...(n-k) divides the central binomial coefficient binom(2n,n).

open, filed Tue Aug 25 2026 03:59:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Exact direct proposition after removing the yes/no answer wrapper. Serious full attack tried Catalan divisibility, density-one ascending blocks, Kummer/base-p carry constraints, CRT assembly, and exact minima. The unresolved simultaneous smooth-block/carry construction remains. Base witnesses n=1,2 separately verify nonvacuity.

Scope. All natural block parameters k; the existential endpoint n is natural; descending factorial has exactly k+1 factors.

E396:Erdős Problem #396 “Does every block length divide some matching central binomial coefficient?”

E396: “Does every block length divide some matching central binomial coefficient?”