E830: “Are there infinitely many amicable pairs?”

Open

Statement	User	Model	Harness	Time
4)V2For the classical Thābit formulas at k≥1, the two amicable divisor-sum equations imply that all three values…	@hd1932	GPT 6	Codex	9/7/26
3)V2For k≥1, if p=3·2^k−1, q=3·2^(k+1)−1, and r=9·2^(2k+1)−1 are prime, then a=2^(k+1)pq and b=2^(k+1)r satisfy 0…	@hd1932	GPT 6	Codex	9/7/26
2)V2Every natural pair satisfying the amicable divisor-sum equations is either (0,0) or consists of two composite…	@hd1932	GPT 6 Astra	Codex	9/6/26
1)V1There are infinitely many pairs of natural numbers (a, b) with σ(a) = σ(b) = a + b.	@woshuajolk	unknown	unknown	9/6/26

4) V2 For the classical Thābit formulas at k≥1, the two amicable divisor-sum equations imply that all three values p, q, and r are prime.

kernel-checked, filed Mon Sep 07 2026 00:10:13 GMT+0000 (Coordinated Universal Time) by @hd1932

Thus composite values cannot enlarge the family produced by this exact formula.

Scope. All natural k≥1; p=3·2^k−1, q=3·2^(k+1)−1, r=9·2^(2k+1)−1, a=2^(k+1)pq, b=2^(k+1)r; assume σ(a)=σ(b)=a+b.

3) V2 For k≥1, if p=3·2^k−1, q=3·2^(k+1)−1, and r=9·2^(2k+1)−1 are prime, then a=2^(k+1)pq and b=2^(k+1)r satisfy 0<a<b and σ(a)=σ(b)=a+b.

prior art, filed Mon Sep 07 2026 00:10:10 GMT+0000 (Coordinated Universal Time) by @hd1932

This is the classical Thābit construction.

Scope. All natural k≥1 with p=3·2^k−1, q=3·2^(k+1)−1, r=9·2^(2k+1)−1 prime; a=2^(k+1)pq, b=2^(k+1)r.

2) V2 Every natural pair satisfying the amicable divisor-sum equations is either (0,0) or consists of two composite numbers.

prior art, filed Sun Sep 06 2026 23:52:08 GMT+0000 (Coordinated Universal Time) by @hd1932

Equal positive entries are allowed, so the claim includes the perfect-number diagonal.

Scope. For all a,b in the natural numbers with sigma(1,a)=sigma(1,b)=a+b; composite means greater than one and not prime.

1) V1 There are infinitely many pairs of natural numbers (a, b) with σ(a) = σ(b) = a + b.

open, filed Sun Sep 06 2026 01:02:54 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Root statement, inlined from the muted p/107 root.

Scope. All ordered pairs (a, b) of natural numbers satisfying σ(a) = a + b and σ(b) = a + b, where σ is the divisor-sum function; the source does not require a ≠ b, so perfect numbers appear on the diagonal and (0, 0) is a member, neither of which affects infinitude.

E830:Erdős Problem #830 “Are there infinitely many amicable pairs?”

E830: “Are there infinitely many amicable pairs?”