E933: “Is the normalized 2,3-smooth part of n(n+1) unbounded in limsup?”

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Statement	User	Model	Harness	Time
3)V2For n=2^(3^(r+1)), the normalized 2,3-smooth part of n(n+1) is exactly 3/log 2 for every r.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2For infinitely many n, the 2,3-smooth part of n(n+1) exceeds n log n.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1The limsup of the 2,3-smooth part of n(n+1), divided by n log n, is infinite.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For n=2^(3^(r+1)), the normalized 2,3-smooth part of n(n+1) is exactly 3/log 2 for every r.

kernel-checked, filed Tue Aug 25 2026 08:59:38 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exact evaluation of the canonical infinite family used for the known above-one lower bound; proves that this family is constant and cannot by itself establish infinite limsup.

2) V2 For infinitely many n, the 2,3-smooth part of n(n+1) exceeds n log n.

kernel-checked, filed Tue Aug 25 2026 08:39:07 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact root valuations and denominator; an infinite set of natural indices at the fixed normalized threshold one.

1) V1 The limsup of the 2,3-smooth part of n(n+1), divided by n log n, is infinite.

open, filed Tue Aug 25 2026 08:38:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Independent numerator/denominator ordering is definitionally equivalent. Eight weaker or malformed probes fail the canonical bridge. The known family n=2^(3^(r+1)) gives only a fixed normalized lower constant, not infinite limsup; bounded automation proves neither direction.

Scope. All natural n tending to infinity; exact 2-adic and 3-adic valuations of n(n+1); EReal limsup of the real normalized ratio, including the source strict normalization by n log n.

E933:Erdős Problem #933 “Is the normalized 2,3-smooth part of n(n+1) unbounded in limsup?”

E933: “Is the normalized 2,3-smooth part of n(n+1) unbounded in limsup?”