Common Sum of 3 Distinct Amicable Pairs

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Theorem

The integer $64 \, 795 \, 852 \, 800$ is the sum of $3$ distinct amicable pairs:

$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$
$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$
$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$

all of them odd.


Proof

We have that:


From $29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ are amicable:

$29 \, 912 \, 035 \, 725$ and $34 \, 883 \, 817 \, 075$ are an odd amicable pair:

$\sigma \left({29 \, 912 \, 035 \, 725}\right) = \sigma \left({34 \, 883 \, 817 \, 075}\right) = 64 \, 795 \, 852 \, 800 = 29 \, 912 \, 035 \, 725 + 34 \, 883 \, 817 \, 075$


From $31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$ are amicable:

$31 \, 695 \, 652 \, 275$ and $33 \, 100 \, 200 \, 525$ are an odd amicable pair:

$\sigma \left({31 \, 695 \, 652 \, 275}\right) = \sigma \left({33 \, 100 \, 200 \, 525}\right) = 64 \, 795 \, 852 \, 800 = 31 \, 695 \, 652 \, 275 + 33 \, 100 \, 200 \, 525$


From $32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$ are amicable:

$32 \, 129 \, 958 \, 525$ and $32 \, 665 \, 894 \, 275$ are an odd amicable pair:

$\sigma \left({32 \, 129 \, 958 \, 525}\right) = \sigma \left({32 \, 665 \, 894 \, 275}\right) = 64 \, 795 \, 852 \, 800 = 32 \, 129 \, 958 \, 525 + 32 \, 665 \, 894 \, 275$


$\blacksquare$


Sources