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Level A · Machine-checkable Hard Number theory P-solitary-number

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.

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 2026-09-28}
}

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The problem

The question

is_ten_solitary. Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is , with abundancy index .

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 . It is unknown whether any club is infinite.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SolitaryNumber (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 is_ten_solitary : answer(sorry) ↔ IsSolitary 10
theorem infinite_club_exists :
    answer(sorry) ↔ ∃ 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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.