Common Sum of 3 Distinct Amicable Pairs

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:

$\map {\sigma_1} {29 \, 912 \, 035 \, 725} = \map {\sigma_1} {34 \, 883 \, 817 \, 075} = 64 \, 795 \, 852 \, 800 = 29 \, 912 \, 035 \, 725 + 34 \, 883 \, 817 \, 075$

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:

$\map {\sigma_1} {31 \, 695 \, 652 \, 275} = \map {\sigma_1} {33 \, 100 \, 200 \, 525} = 64 \, 795 \, 852 \, 800 = 31 \, 695 \, 652 \, 275 + 33 \, 100 \, 200 \, 525$

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:

$\map {\sigma_1} {32 \, 129 \, 958 \, 525} = \map {\sigma_1} {32 \, 665 \, 894 \, 275} = 64 \, 795 \, 852 \, 800 = 32 \, 129 \, 958 \, 525 + 32 \, 665 \, 894 \, 275$

$\blacksquare$

Sources

• Oct. 1993: David Moews and Paul C. Moews: A Search for Aliquot Cycles and Amicable Pairs (Math. Comp. Vol. 61, no. 204: pp. 935 – 938)  www.jstor.org/stable/2153265

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $64,795,852,800$