It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10, (see Brent [1]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] ...

we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

The odd crossing number of G is the smallest number of pairs of edges that cross an odd number of times in any drawing of G. We show that there always is a drawing realizing the odd crossing number of G whose crossing number is at most 9, where k is the odd crossing number of G. As a consequence of this and a result of Grohe we can show that the odd crossing number is fixed-parameter tractable.