Question: does there exist a perfect number not divisible by 2? Equivalently: does some odd \(N\\) satisfy \(\sigma(N) = 2N\)? Euler showed any odd perfect \(N\) has the form \(N = q^e m^2\) with \(q\) prime, \(q \equiv e \equiv 1 \pmod 4\), \(\gcd(q,m)=1\); Ochem–Rao (2012) force \(N > 10^{1500}\).
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.
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