[#P38] Nonexistence of odd perfect numbers

Problem. There is no odd integer $N>0$ whose sum of positive divisors satisfies $\sigma(N)=2N$.

Divisor-pair diagram for a candidate perfect number.

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Definitions and notation

1Context

Even perfect numbers have a classical classification. No odd example is known, and strong necessary conditions have been proved for any hypothetical example.

2Definitions

Definition 1 (A proper divisor of N). A proper divisor of N is a positive divisor smaller than N.

Definition 2 (A perfect number). A perfect number is a positive integer equal to the sum of its positive proper divisors; equivalently, its positive-divisor sum is 2N.

3What counts as a solution

Prove that every perfect number is even, or exhibit an odd positive integer and verify exactly that the sum of its proper divisors equals the integer.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Any odd perfect number exceeds 10^1500, has at least 101 prime factors counted with multiplicity, and obeys further severe factorization restrictions. Exact unresolved remainder: Prove that no odd positive integer N satisfies sigma(N)=2N.^[2][1]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Any odd perfect number exceeds 10^1500, has at least 101 prime factors counted with multiplicity, and obeys further severe factorization restrictions. Exact unresolved remainder: Prove that no odd positive integer N satisfies sigma(N)=2N.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Any odd perfect number exceeds 10^1500, has at least 101 prime factors counted with multiplicity, and obeys further severe factorization restrictions. Exact unresolved remainder: Prove that no odd positive integer N satisfies sigma(N)=2N.^[2][1] Relevance to this problem For Nonexistence of odd perfect numbers, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Any odd perfect number exceeds 10^1500, has at least 101 prime factors counted with multiplicity, and obeys further severe factorization restrictions. Exact unresolved remainder: Prove that no odd positive integer N satisfies sigma(N)=2N. Open full record →
Prove that every perfect number is even, or exhibit an odd positive integer and verify exactly that the sum of its proper divisors equals the integer.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that every perfect number is even, or exhibit an odd positive integer and verify exactly that the sum of its proper divisors equals the integer.^[1] Relevance to this problem For Nonexistence of odd perfect numbers, record nonexistence-of-odd-perfect-numbers-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that every perfect number is even, or exhibit an odd positive integer and verify exactly that the sum of its proper divisors equals the integer. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that every perfect number is even, or exhibit an odd positive integer and verify exactly that the sum of its proper divisors equals the integer. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited American Mathematical Society article says that the existence of an odd perfect number is unknown. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Any candidate must satisfy many published lower bounds and factorization constraints; consult the current literature before reporting a search result.

Recorded example 1. The integers 6 and 28 are even perfect numbers: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.

Computational notes

Excluding odd candidates through a finite bound improves a lower bound and does not prove nonexistence.

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Nonexistence of odd perfect numbers.” TheoremDB. P38. Problem statement; statement text SHA-256 b4cb01f59ed784352c9c4920e3768b7a5baf6be3fc3b0d3f1e314f3a99dc3291. https://theoremdb.org/statement/?ref=P38

BibTeX

@misc{theoremdb-problem-b4cb01f59ed784352c9c4920e3768b7a5baf6be3fc3b0d3f1e314f3a99dc3291,
  title = {{Nonexistence of odd perfect numbers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b4cb01f59ed784352c9c4920e3768b7a5baf6be3fc3b0d3f1e314f3a99dc3291},
  url = {https://theoremdb.org/statement/?ref=P38}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathvoices.ams.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. American Mathematical Society Math Voices, Tony's Take October 2024: Math in the Media, checked 2026-08-01. American Mathematical Society Math Voices discussion of perfect numbers. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited American Mathematical Society article says that the existence of an odd perfect number is unknown. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Odd perfect numbers item.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States that the existence of an odd perfect number remains unknown and points to the current size restrictions.Source named by the research packet.
Pascal Ochem and Michaël Rao, “Odd perfect numbers are greater than $10^{1500}$”. Mathematics of Computation 81(279) (2012), 1869-1877. DOI 10.1090/S0025-5718-2012-02563-4. Abstract and main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Current strong size and factorization restrictions.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P38] Nonexistence of odd perfect numbers

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