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Level A · Machine-checkable Hard Number theory P-idoneal-completeness

Idoneal numbers completeness conjecture

Idoneal numbers completeness conjecture.

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-idoneal-completeness,
  title        = {Idoneal numbers completeness conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/idoneal-completeness}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Idoneal numbers completeness conjecture.

An integer is idoneal if every integer that can be expressed in exactly one way (up to order and signs) as with gcd(x, Dy)=1 is a prime power or twice a prime power.

The Idoneal Numbers Completeness Conjecture asserts that the following list of 65 numbers is complete: 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48, 57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232, 240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.IdonealCompleteness. answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem idoneal_numbers_completeness :
    answer(sorry) ↔
      ∀ n : ℕ, IsIdoneal n → n ∈ knownIdonealNumbers

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • 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.