Existence of odd perfect numbers

A perfect number equals the sum of its proper divisors (e.g. 6 = 1 + 2 + 3). All known perfect numbers are even. Whether any odd perfect number exists is one of the oldest open problems in mathematics, dating back to Euclid.

Lower bounds have grown — any odd perfect number must exceed 10^1500 and satisfy strict structural constraints — but no proof of non-existence is known.