Odd Perfect Numbers

Question: does there exist a perfect number not divisible by 2? Equivalently: does some odd N satisfy σ(N) = 2N?

Euclid–Euler (circa 300 BC / 1749): even perfect numbers are exactly \(2^{p−1}\)(\(2^{p}\)−1) with \(2^{p}\)−1 prime — so the entire question is the odd case, open since Nicomachus catalogued them c. 100 AD. Exhaustive constraints accumulate: any odd perfect number exceeds \(10^{1500}\) (Ochem–Rao), has at least nine distinct prime factors, a prime factor exceeding \(10^{8}\), and must take one of finitely many Eulerian forms N = p^α m². Each new paper adds a constraint; none closes.