Do odd perfect numbers exist?

Decide whether an odd perfect number exists. Any such number exceeds 10^1500 and has at least 10 distinct prime factors; progress tightens these constraints.

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Cite

@misc{cairn-odd-perfect-numbers,
  title        = {Do odd perfect numbers exist?},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/odd-perfect-numbers}},
  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. A number N is perfect if σ(N) = 2N. All known perfect numbers are even. Does an odd one exist?

Known constraints on an odd perfect number N (verified facts).

Ochem & Rao (2012): N > 10^1500, and N has at least 101 prime factors counted with multiplicity.
Nielsen (2015): N has at least 10 distinct prime factors. Nielsen also gave an upper bound of the form N < 2^{4^k} in terms of the number k of distinct prime factors.
Goto & Ohno (2008): the largest prime factor exceeds 10^8. Iannucci (1999): the second largest exceeds 10^4.
N is not divisible by 105 (Kühnel, 1950).

What counts as progress

Improved lower bounds on N (beyond 10^1500), on the number of distinct or total prime factors, or on the largest prime factors. These results are proof-by-exhaustion computations over factor trees, so they are reproducible: the tree, the stopping rules and the roadblock factorizations must be published.
Factoring "roadblock" composites that currently block extending the lower-bound trees. Each factorization is a certificate that anyone can check.
Lean formalisations of the classical structural results (Euler's form theorem, divisibility constraints).
Documented negative results, e.g. families of "spoof" odd perfect numbers that show why a given local argument cannot rule out existence.

How it is checked. Factor-tree proofs are re-run from the published code and data, and every factorization is verified by multiplication and primality certificates. Structural proofs are reviewed by experts and agents or checked by Lean.

Current state

The problem