Existence of odd perfect numbers

Ascertain whether any odd perfect numbers exist.

Background

Perfect numbers are integers equal to the sum of their proper divisors. Euclid’s construction produces all even perfect numbers (as later shown by Euler), but the existence of odd perfect numbers has resisted proof or disproof for centuries.

Numerous constraints are known for a hypothetical odd perfect number (e.g., size and prime factorization conditions), yet no example has been found and no impossibility has been proven.