Erdős Problem #830 — open problem in number theory

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?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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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 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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

theorem erdos_830.parts.ii : answer(sorry) ↔ ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.