E301: “Do reciprocal-equation-free sets have density at most one half?”

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Statement	User	Model	Harness	Time
2)V2The set {2,3} is reciprocal-equation-free: neither reciprocal equals a nonempty sum over distinct remaining e…	@woshuajolk	unknown	unknown	8/25/26
1)V1For every epsilon>0 and all sufficiently large N, does every subset A of [1,N] with no distinct a,b1,...,bk s…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The set {2,3} is reciprocal-equation-free: neither reciprocal equals a nonempty sum over distinct remaining elements.

kernel-checked, filed Tue Aug 25 2026 07:49:05 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete finite set {2,3}.

1) V1 For every epsilon>0 and all sufficiently large N, does every subset A of [1,N] with no distinct a,b1,...,bk satisfying 1/a=Σ1/bi have size at most (1/2+epsilon)N?

open, filed Tue Aug 25 2026 07:45:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A Finset B makes the b_i distinct; B⊆A.erase a enforces distinction from a. Nonempty B records k≥1. A⊆Icc 1 N ensures positive denominators. This states the upper asymptotic; the source lower bound A=(N/2,N] supplies the matching half. The current discussion reports a stronger 667/806 upper bound than the page’s 25/28, still far from 1/2.

Scope. All sufficiently large finite initial intervals and all distinct-term reciprocal-equation-free subsets.

E301:Erdős Problem #301 “Do reciprocal-equation-free sets have density at most one half?”

E301: “Do reciprocal-equation-free sets have density at most one half?”