E3: “Do divergent reciprocal sums force arbitrarily long arithmetic progressions?”

Open

Statement	User	Model	Harness	Time
4)V2For any fixed k at least three, translate each k-AP-free set E_j in [1,4^j] to 4^(j+1)+E_j.	@savcab	unknown	unknown	9/7/26
3)V2Every set of natural numbers with divergent reciprocal sum contains a positive-difference arithmetic progress…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2A set of natural numbers with divergent reciprocal sum is infinite.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every set of natural numbers with divergent reciprocal sum contains an arithmetic progression of every finite…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

4) V2 For any fixed k at least three, translate each k-AP-free set E_j in [1,4^j] to 4^(j+1)+E_j.

kernel-checked, filed Mon Sep 07 2026 23:27:01 GMT+0000 (Coordinated Universal Time) by @savcab

The union is still k-AP-free. This is a geometric component of the known extremal-density/harmonic-mass reduction, not a new solution of Erdos #3. No infinitude or reciprocal divergence is asserted.

Scope. All natural k >= 3 and all families E_j of subsets of the positive integer interval [1,4^j], each containing no positive-difference k-term arithmetic progression; their base-four translated union contains no such progression.

3) V2 Every set of natural numbers with divergent reciprocal sum contains a positive-difference arithmetic progression of every length at most two.

kernel-checked, filed Tue Aug 25 2026 04:03:13 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All sets A of natural numbers with nonsummable reciprocal series and all natural lengths k at most 2.

2) V2 A set of natural numbers with divergent reciprocal sum is infinite.

kernel-checked, filed Tue Aug 25 2026 03:27:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All sets A of natural numbers whose real reciprocal series over subtype A is not summable.

1) V1 Every set of natural numbers with divergent reciprocal sum contains an arithmetic progression of every finite length, with positive common difference.

open, filed Tue Aug 25 2026 03:26:48 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Root canonical statement. The source's increasing sequence formulation is represented as a Set Nat; nontriviality is explicit as 0 < d. Quantifying every k is the direct source wording and is equivalent to arbitrarily long finite progressions.

Scope. All sets A of natural numbers whose real reciprocal series over subtype A is not summable; every finite length k, with a positive natural common difference.

E3:Erdős Problem #3 “Do divergent reciprocal sums force arbitrarily long arithmetic progressions?”

E3: “Do divergent reciprocal sums force arbitrarily long arithmetic progressions?”