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Odd perfect numbers
Are there any odd perfect numbers? 
It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.
Any odd perfect number N must satisfy the following conditions:
 N > 10^{1500}, result published in 2012.
 N is of the form
 where:
 q, p_{1}, ..., p_{k} are distinct primes (Euler).
 q ≡ α ≡ 1 (mod 4) (Euler).
 The smallest prime factor of N is less than (2k + 8) / 3.
 Either q^{α} > 10^{62}, or p_{ j}^{2ej} > 10^{62} for some j.
 N < 2^{4k+1}.^{[11]}
 The largest prime factor of N is greater than 10^{8}.
 The second largest prime factor is greater than 10^{4}, and the third largest prime factor is greater than 100.^{[13]}^{[14]}
 N has at least 101 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.
In 1888, Sylvester stated:
...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.
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