There exist no odd perfect numbers

Proof:
Suppose sigma(n)=2n [n perfect]
sigma(pn)=2n(p+1) [if hcf(n,p)=1]

p+1==0 [mod 2] [if p odd]
=> sigma(pn)==0 [mod 4]

Let p->99999... [Euclid]
sigma(pn)/pn->2

=> pn==0 [mod 2]
=> n==0 [mod 2] [since p odd]

Copyright 1997 James Wanless