Practical numbers — open problem in number theory

Practical numbers

Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number. - Hal M. Switkay, Jan 28 2023

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-oeis-5153,
  title        = {Practical numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-5153}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number.

Hal M. Switkay, Jan 28 2023

A positive integer $n$ is called a practical number (or panarithmic number) if every positive integer $m \leq n$ can be represented as a sum of distinct divisors of $n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«5153».

theorem conjecture (n : ℕ) (hn : 3 ≤ n) (hodd : Odd n) :
    ∃ p q : ℕ, p.Prime ∧ A q ∧ n = p + q

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

A005153

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.