amicablenumbers

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]