Definition:Amicable Pair/Definition 1

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Definition

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.

$m$ and $n$ are an amicable pair if and only if:

the aliquot sum of $m$ is equal to $n$

and:

the aliquot sum of $n$ is equal to $m$.


Examples

$220$ and $284$

$220$ and $284$ are the smallest amicable pair:

$\map \sigma {220} = \map \sigma {284} = 504 = 220 + 284$


$1184$ and $1210$

$1184$ and $1210$ are the $2$nd amicable pair:

$\map \sigma {1184} = \map \sigma {1210} = 2394 = 1184 + 1210$


$2620$ and $2924$

$2620$ and $2924$ are the $3$rd amicable pair:

$\map \sigma {2620} = \map \sigma {2924} = 5544 = 2620 + 2924$


$5020$ and $5564$

$5020$ and $5564$ are the $4$th amicable pair:

$\map \sigma {5020} = \map \sigma {5564} = 10 \, 584 = 5020 + 5564$


$6232$ and $6368$

$6232$ and $6368$ are the $5$th amicable pair:

$\sigma \left({6232}\right) = \sigma \left({6368}\right) = 12 \, 600 = 6232 + 6368$


$10 \, 744$ and $10 \, 856$

$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:

$\sigma \left({10 \, 744}\right) = \sigma \left({10 \, 856}\right) = 21 \, 600 = 10 \, 744 + 10 \, 856$


$12 \, 285$ and $14 \, 595$

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

$\map \sigma {12 \, 285} = \map \sigma {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$


$17 \, 296$ and $18 \, 416$

$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:

$\map \sigma {17 \, 296} = \map \sigma {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$


Sequence

The sequence of amicable pairs begins:

$\tuple {220, 284}, \tuple {1184, 1210}, \tuple {2620, 2924}, \tuple {5020, 5564}, \tuple {6232, 6368}, \tuple {10744, 10856}, \tuple {12285, 14595}, \tuple {17296, 18416}, \tuple {63020, 76084}$


Also see

  • Results about amicable pairs can be found here.


Historical Note

Amicable pairs are on record as having been studied by the Pythagoreans.

The first mathematician to explore amicable pairs systematically was Leonhard Paul Euler.

He published a list of $64$ examples.

There are now over $40 \, 000$ amicable pairs now known, including all such pairs where the smaller pair is under $1 \, 000 \, 000$.

Techniques for generating new amicable pairs from existing ones strongly suggest that there is an infinite number of them, but this still has to be rigorously proven.


Sources