Erdős (Erdos) Problem #830: status in 2026

Erdős Problem #830

Erdos Problem 830, Part 1 We say that a,b∈ ℕ are an amicable pair if σ(a)=σ(b)=a+b. Are there infinitely many amicable pairs?

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #830 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #830 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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Cite

@misc{cairn-erdos-830,
  title        = {Erdős Problem #830},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-830}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_830.parts.i. Erdos Problem 830, Part 1 We say that $a, b \in N$ are an amicable pair if $σ (a) = σ (b) = a + b$ . Are there infinitely many amicable pairs?

erdos_830.parts.ii. Erdos Problem 830, Part 2 We say that $a, b \in N$ are an amicable pair if $σ (a) = σ (b) = a + b$ . If $A (x)$ counts the number of amicable $1 \leq a \leq b \leq x$ then is it true that $A (x) > x^{1 - o (1)} ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«830» (2 statements).

theorem erdos_830.parts.i : {(a, b) | IsAmicable a b}.Infinite

theorem erdos_830.parts.ii : ∃ o : ℝ → ℝ, o =o[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop,
    x ^ (1 - o x) < A x

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/830. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/830
[Er55b] Erdős, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Po15] Pomerance, Carl, On amicable numbers. (2015), 321-327.
[Po81] Pomerance, Carl, On the distribution of amicable numbers. II. J. Reine Angew. Math. (1981), 183-188.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.