[#P38] Nonexistence of odd perfect numbers
Problem. There is no odd integer \(N>0\) whose sum of positive divisors satisfies \(\sigma(N)=2N\).
1Context
Even perfect numbers have a classical classification. No odd example is known, and strong necessary conditions have been proved for any hypothetical example.
2Problem setup
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.
Remark 1. Even perfect numbers have a classical classification. No odd example is known, and strong necessary conditions have been proved for any hypothetical example.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Nonexistence of odd perfect numbers,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/nonexistence-of-odd-perfect-numbersThis page as plain text: nonexistence-of-odd-perfect-numbers.md
This problem includes 2 records joined by 2 typed links, sourced from mathvoices.ams.org[1], current as of July 31, 2026.
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.