What are perfect numbers as discussed in Euclid’s Elements?

In his work Elements, the Greek mathematician Euclid defined a perfect number as an integer that is equal to the sum of its "parts," which refers to its proper divisors.1

Euclid provided a significant mathematical "recipe" for generating these numbers, which marks the earliest extant result concerning them. He proved that if a series of numbers beginning from a unit (1) is set out in a "double proportion" (1, 2, 4, 8, ...) until their sum results in a prime number, then the product of that sum and the last number in the series will be a perfect number.21

In modern notation, Euclid’s formula states that if $2^n-1$ is prime, then $2^{n-1}(2^n-1)$ is a perfect number. For example, because $1+2+4=7$ (a prime), the product $4\times 7=28$ is a perfect number. While Euclid’s formula identifies even perfect numbers, it remains unknown whether any odd perfect numbers exist.3421