E891: “Prime-factor-rich integers in primorial-length intervals.”

Open

Statement	User	Model	Harness	Time
2)V2Every sufficiently late unit interval contains an integer with more than zero distinct prime factors.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every k≥2, every sufficiently late half-open interval of length equal to the product of the first k prime…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every sufficiently late unit interval contains an integer with more than zero distinct prime factors.

kernel-checked, filed Tue Aug 25 2026 05:48:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The k=0 calibration, where the empty product gives interval length one.

1) V1 For every k≥2, every sufficiently late half-open interval of length equal to the product of the first k primes contains an integer with more than k distinct prime factors.

open, filed Tue Aug 25 2026 05:46:53 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical concrete right-hand proposition. The interval is half-open, the first-k-prime product uses indices 0 through k-1, and `ω` counts distinct prime factors rather than multiplicity.

Scope. Natural k,n,m; zero-indexed nth-prime enumeration begins with 2; ω counts distinct prime factors.

E891:Erdős Problem #891 “Prime-factor-rich integers in primorial-length intervals.”

E891: “Prime-factor-rich integers in primorial-length intervals.”