Amicable Numbers
All amicable pairs’ sum is even
Proof:
Suppose sigma(n)=n+m=sigma(m) [m,n amicable]
sigma(n)+sigma(m)=2(n+m)
sigma(pn)+sigma(pm)=2(n+m)(p+1) [if hcf(n,p)=1]
p+1==0 [mod 2] [if p odd]
=> sigma(pn)+sigma(pm)==0 [mod 4]
Let p->99999... [Euclid]
[sigma(pn)+sigma(pm)]/(pn+pm)->2
=> p(n+m)==0 [mod 2]
=> n+m==0 [mod 2] [since p odd]