Odd perfect numbers

As of 2026 no odd perfect number has been found, and whether one can exist remains open. Instead of a direct proof of nonexistence, a web of necessary conditions has been established that any hypothetical odd perfect number $N$ must satisfy: by Euler's theorem $N = q^k m^2$ where $q$ is a prime with $q \equiv k \equiv 1 \pmod{4}$ and $\gcd(q, m) = 1$; $N > 10^{1500}$ and its total number of prime factors counted with multiplicity is $\Omega(N) \ge 101$ (Ochem–Rao, 2012); $N$ has at least $\omega(N) \ge 10$ distinct prime factors, and $\omega(N) \ge 15$ if $3 \nmid N$ (Nielsen, 2015); its largest prime factor exceeds $10^8$ (Goto–Ohno, 2008), its second largest exceeds $10^4$, and its largest prime-power component exceeds $10^{62}$. However, as Sylvester and later researchers noted, cyclotomic factorizations of $\sigma(p^a) = \prod_{d \mid (a+1), d>1} \Phi_d(p)$ apply equally to 'spoof' factorizations where a prime factor is replaced by a composite quasi-prime, and since Descartes's spoof $198585576189$ exists, purely local divisibility chains face a spoof barrier unless they use a deeper global property of primes.