Odd Perfect Number Conjecture

A perfect number equals the sum of its proper divisors, that is, $\sigma(n)=2n$. The even ones are completely understood: by results going back to Euclid and completed by Euler, they are exactly the numbers $2^{p-1}(2^p-1)$ with $2^p-1$ prime. Whether an odd perfect number exists is a question of genuine antiquity — so old that no first proposer or date can be assigned reliably — and the conjecture is that none does. Euler began its modern study, showing that an odd perfect number would have to take the rigid form $p^{a}m^{2}$ with $p$ prime and $p\equiv a\equiv1\pmod 4$ [Euler1849Perfect].

Modern work has accumulated severe necessary conditions on a hypothetical example. Nielsen proved that an odd perfect number must have at least nine distinct prime factors [Nielsen2007OddPerfect], and Ochem and Rao showed that it must exceed $10^{1500}$ [OchemRao2012OddPerfect]; a web of congruence restrictions constrains its shape further.

None of these constraints has produced a contradiction, and no example has been found. The problem remains open: a resolution requires either exhibiting an odd $n$ with $\sigma(n)=2n$ or proving that the ever-tightening conditions are jointly unsatisfiable.