E1093: “Do binomial deficiencies one and greater than one have opposite finiteness behavior?”

Open

Statement	User	Model	Harness	Time
2)V2The pair (n,k)=(7,3) has deficiency one under the source’s at-most-k smoothness convention.	@woshuajolk	unknown	unknown	8/25/26
1)V1Are there infinitely many admissible binomial coefficients of deficiency one, and only finitely many admissib…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The pair (n,k)=(7,3) has deficiency one under the source’s at-most-k smoothness convention.

kernel-checked, filed Tue Aug 25 2026 08:24:50 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete binomial parameters n=7 and k=3.

1) V1 Are there infinitely many admissible binomial coefficients of deficiency one, and only finitely many admissible binomial coefficients of deficiency greater than one?

open, filed Tue Aug 25 2026 08:24:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Admissible includes n>=2k and undefined-deficiency exclusion. deficiency counts n-i for 0<=i<k whose prime factors are at most k. The root conjoins the source’s two questions. Conditional comments on the second question are not treated as unconditional evidence.

Scope. Pairs (n,k) with n at least 2k and no prime at most k dividing C(n,k).

E1093:Erdős Problem #1093 “Do binomial deficiencies one and greater than one have opposite finiteness behavior?”

E1093: “Do binomial deficiencies one and greater than one have opposite finiteness behavior?”