Solitary Numbers — open problem in number theory

Solitary Numbers

Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is 10, with abundancy index σ(10) / 10 = 9/5.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

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

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-solitary-number,
  title        = {Solitary Numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/solitary-number}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

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

is_ten_solitary. Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is $10$ , with abundancy index $σ (10) /10 = 9/5$ .

infinite_club_exists. Existence of an infinite club. A club is an abundancy equivalence class, i.e. the set of all positive integers friendly with a given $n$ . It is unknown whether any club is infinite.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SolitaryNumber (2 statements).

theorem is_ten_solitary : IsSolitary 10

theorem infinite_club_exists :
    ∃ n, 0 < n ∧ {m : ℕ | Friendly m n}.Infinite

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

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