Sylvester: an odd perfect number has at least five distinct prime divisors

Sylvester (1888). If $N$ is odd and perfect, then $\omega(N) \ge 5$: $N$ has at least five distinct prime divisors.

The proof refines the abundancy estimate $\sigma(N)/N < \prod_{p \mid N} p/(p-1)$ by a case analysis over the possible small prime supports, using the multiplicativity of $\sigma$ and the constraints coming from Euler's form. Sylvester proved in the same work the stronger bound $\omega(N) \ge 8$ under the additional hypothesis $3 \nmid N$; the unconditional bound has since been improved to $\omega(N) \ge 8$ (Chein 1979, Hagis 1980), $\ge 9$ (Nielsen 2007) and $\ge 10$ (Nielsen 2015). Any of those stronger statements also settles this milestone.

Formalized with $\omega(N)$ as N.primeFactors.card.