E889: “Do arbitrarily many new prime factors appear in consecutive-integer prefixes?”

Open

Statement	User	Model	Harness	Time
2)V2At offset zero, v(n,0) is exactly the number of distinct prime factors of n, and this term lower-bounds v₀(n).	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Let v(n,k) count prime factors of n+k dividing none of n,...,n+k-1, and v₀(n)=sup_k v(n,k).	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 At offset zero, v(n,0) is exactly the number of distinct prime factors of n, and this term lower-bounds v₀(n).

kernel-checked, filed Tue Aug 25 2026 06:40:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every natural n.

1) V1 Let v(n,k) count prime factors of n+k dividing none of n,...,n+k-1, and v₀(n)=sup_k v(n,k).

open, filed Tue Aug 25 2026 06:40:35 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Does v₀(n) tend to infinity?

The finite filter exactly counts distinct prime factors of n+k absent from all earlier n+i. ENat iSup faithfully models the source maximum without assuming a priori boundedness. Convergence is to top in ENat.

Scope. All natural starting values n and all nonnegative offsets k.

E889:Erdős Problem #889 “Do arbitrarily many new prime factors appear in consecutive-integer prefixes?”

E889: “Do arbitrarily many new prime factors appear in consecutive-integer prefixes?”