E873: “Do consecutive LCM windows in every increasing sequence have subpower counting growth?”

Open

Statement	User	Model	Harness	Time
3)V2For every positive strictly increasing sequence and every positive natural threshold N, fewer than N one-term…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2For window length zero and threshold two, every start index is counted, so the extended count is infinite.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every strictly increasing positive sequence of natural numbers and every positive epsilon, some fixed win…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For every positive strictly increasing sequence and every positive natural threshold N, fewer than N one-term windows have LCM below N.

kernel-checked, filed Tue Aug 25 2026 08:55:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exact k=1 counting bound at all positive natural thresholds; uses ENat encard, matching the root counting semantics.

2) V2 For window length zero and threshold two, every start index is counted, so the extended count is infinite.

kernel-checked, filed Tue Aug 25 2026 08:20:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural sequences a; the exact root block-LCM and extended-cardinality definitions at k=0 and X=2.

1) V1 For every strictly increasing positive sequence of natural numbers and every positive epsilon, some fixed window length makes the number of consecutive blocks with LCM below X less than X to the epsilon for every positive X.

open, filed Tue Aug 25 2026 08:20:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The first draft used Set.ncard and was rejected locally because k=0 would turn an infinite count into zero. The final root uses encard/ℕ∞ exactly to close that exploit. Independent transcription is definitionally equal; seven malformed probes fail; k=0 is witnessed to give infinite count at X=2.

Scope. All strictly increasing a:ℕ→ℕ with a(0)>0; every real epsilon>0; one natural window length k independent of X; extended-natural cardinality counts all start indices and prevents infinite sets from collapsing to zero.

E873:Erdős Problem #873 “Do consecutive LCM windows in every increasing sequence have subpower counting growth?”

E873: “Do consecutive LCM windows in every increasing sequence have subpower counting growth?”