Sylvester's conjecture on odd perfect numbers

Let $n$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. A positive integer is perfect if

$$\sigma(n)=2n.$$

Sylvester's conjecture. There is no odd perfect number.

The conjecture asserts that every perfect number is even; all currently known perfect numbers are even, while the existence of any odd perfect number remains open. The paper notes that any odd perfect number, if one exists, must exceed $10^{1500}$.