An amicable pair consists of two Integers for which the sum of Proper Divisors (the Divisors excluding the number itself) of one number equals the other. Amicable pairs
are occasionally called Friendly Pairs, although this nomenclature is to be discouraged since
Friendly Pairs are defined by a different, if related, criterion. Symbolically, amicable pairs
satisfy

(1) | |||

(2) |

where is the Restricted Divisor Function or, equivalently,

(3) |

(4) | |||

(5) |

giving Restricted Divisor Functions

(6) | |||

(7) |

The quantity

(8) |

In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056). By 1747, Euler had found 30 pairs, a number which he later extended to 60. There were 390 known as of 1946 (Scott 1946). There are a total of 236 amicable pairs below (Cohen 1970), 1427 below (te Riele 1986), 3340 less than (Moews and Moews 1993), 4316 less than (Moews and Moews), and 5001 less than (Moews and Moews).

The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (Sloane's A002025 and A002046). An exhaustive tabulation is maintained by D. Moews.

Let an amicable pair be denoted with . is called a regular amicable pair of type if

(9) |

(10) |

(11) |

(12) |

(13) |

te Riele (1986) also found 37 pairs of amicable pairs having the same Pair Sum. The first such pair is (609928, 686072)
and (643336, 652664), which has the Pair Sum

(14) |

On October 4, 1997, Mariano Garcia found the largest known amicable pair, each of whose members has 4829
Digits. The new pair is

(15) | |||

(16) |

where

(17) | |||

(18) | |||

(19) | |||

(20) |

, , , and are Prime.

Pomerance (1981) has proved that

(21) |

**References**

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© 1996-9

1999-05-25