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.

References

Nor do we know whether there are any odd perfect numbers at all.

— How did Fermat discover his theorem?  (2502.11165 - Pengelley, 16 Feb 2025) in Subsection “Perfect Numbers”

It remains unknown whether any odd perfect numbers exist.

— Generalizing the Eight Levels Theorem: A Journey to Mersenne Prime Discoveries and New Polynomial Classes  (2404.05772 - Ibrahim, 2024) in Section 15.2 (Connections with Mersenne and Fermat numbers)