E386: “Can one fixed nontrivial binomial coefficient row produce consecutive-prime products infinitely often?”

Open

Statement	User	Model	Harness	Time
2)V2The source example 21 choose 2 equals the product of the first four consecutive primes: 2·3·5·7.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is there a fixed integer k ≥ 2 such that, for infinitely many n with k ≤ n−2, the binomial coefficient n choo…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The source example 21 choose 2 equals the product of the first four consecutive primes: 2·3·5·7.

kernel-checked, filed Tue Aug 25 2026 04:05:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete nontrivial instance n=21, k=2, and prime indices [0,4).

1) V1 Is there a fixed integer k ≥ 2 such that, for infinitely many n with k ≤ n−2, the binomial coefficient n choose k is a product of consecutive primes?

open, filed Tue Aug 25 2026 04:05:08 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The existential-over-fixed-k interpretation follows the formal-conjectures root exactly; the stronger all-k and specialized k=2 variants are deliberately not conflated with it.

Scope. One fixed k and infinitely many natural n; the prime block is an arbitrary half-open interval of the increasing prime sequence.

E386:Erdős Problem #386 “Can one fixed nontrivial binomial coefficient row produce consecutive-prime products infinitely often?”

E386: “Can one fixed nontrivial binomial coefficient row produce consecutive-prime products infinitely often?”