E824: “Are coprime equal-sigma pairs almost quadratic?”

Open

Statement	User	Model	Harness	Time
2)V2The divisor sums satisfy σ(1)=1 and σ(2)=3, and 1 is coprime to 2.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1If h(x) counts 1≤a<b<x with gcd(a,b)=1 and σ(a)=σ(b), is h(x)>x^(2-o(1))?	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The divisor sums satisfy σ(1)=1 and σ(2)=3, and 1 is coprime to 2.

kernel-checked, filed Tue Aug 25 2026 07:27:50 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. A concrete smoke test of the root verifier's divisor-sum and coprimality predicates.

1) V1 If h(x) counts 1≤a<b<x with gcd(a,b)=1 and σ(a)=σ(b), is h(x)>x^(2-o(1))?

open, filed Tue Aug 25 2026 07:25:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Small sigma values and coprimality kernel-check; an independent encoding is definitionally equal; nine content-free bridges fail. Whole attacks cover primitive friendly pairs, multiplicative factor swaps, sieving, sigma-fiber collision energy, maximal-fiber refutations, and count degeneracies.

Scope. The precise conjecture in Erdős problem 824. Sigma is the sum of positive divisors; each unordered pair is counted once through a<b; positivity, strict cutoff, coprimality, and exact sigma equality are explicit.

E824:Erdős Problem #824 “Are coprime equal-sigma pairs almost quadratic?”

E824: “Are coprime equal-sigma pairs almost quadratic?”