Existence of odd perfect numbers

Determine whether any odd perfect numbers exist, i.e., whether there is an odd positive integer equal to the sum of its proper positive divisors.

Background

While discussing connections between Ψ-polynomials, Mersenne numbers, and perfect numbers, the authors note the longstanding open problem concerning odd perfect numbers. They mention known large lower bounds but emphasize that the basic existence question for odd perfect numbers remains unresolved.