# P38: Nonexistence of odd perfect numbers

- ID: `P38`
- Reference: `nonexistence-of-odd-perfect-numbers`
- Page: https://theoremdb.org/statements/P38
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (A proper divisor of N).** A proper divisor of N is a positive divisor smaller than N.
- **Definition (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.** Even perfect numbers have a classical classification. No odd example is known, and strong necessary conditions have been proved for any hypothetical example.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The public status was checked against the cited American Mathematical Society article on 2026-07-22.
- Any candidate must satisfy many published lower bounds and factorization constraints; consult the current literature before reporting a search result.

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

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `nonexistence-of-odd-perfect-numbers`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://mathvoices.ams.org/mathmedia/tonys-take-october-2024/
   - Also cited at Odd perfect numbers item
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1090/S0025-5718-2012-02563-4
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Current strong size and factorization restrictions.