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 (Hoffman 1998, p. 45), although this nomenclature is to be discouraged
since the numbers more commonly known as friendly pairs
are defined by a different, albeit related, criterion. Symbolically, amicable pairs
satisfy
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(1)
|
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(2)
|
where
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(3)
|
is the restricted divisor function. Equivalently, an amicable pair 
satisfies
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(4)
|
where 
is the divisor function. The smallest amicable
pair is (220, 284) which has factorizations
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(5)
|
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(6)
|
giving restricted divisor functions
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(7)
|
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(8)
|
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(9)
|
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(10)
|
The quantity
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(11)
|
in this case, 
,
is called the pair sum. The first few amicable pairs
are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856),
(12285, 14595), (17296, 18416), (63020, 76084), ... (OEIS A002025
and A002046). An exhaustive tabulation is maintained
by D. Moews.
In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), although these results were actually rediscoveries of numbers known to
Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended
to 60. In 1866, 16-year old B. Nicolò I. Paganini found the small
amicable pair (1184, 1210) which had eluded his more illustrious predecessors (Paganini
1866-1867; Dickson 2005, p. 47). There were 390 known amicable pairs as of 1946
(Escott 1946). There are a total of 236 amicable pairs below 
(Cohen 1970), 1427 below 
(te Riele 1986), 3340 less than 
(Moews and Moews 1993ab), 4316 less than 
(Moews and Moews 1996), and 5001 less than
(Moews and
Moews 1996).
Rules for producing amicable pairs include the Thâbit ibn Kurrah rule rediscovered by Fermat and Descartes and extended by Euler to Euler's rule. A further extension not previously noticed was discovered by Borho (1972).
Pomerance (1981) has proved that
![[amicable numbers <=n]<ne^(-[ln(n)]^(1/3))](/images/equations/AmicablePair/NumberedEquation4.svg) |
(12)
|
for large enough 
(Guy 1994). No nonfinite lower bound has been proven.
Let an amicable pair be denoted 
, and take 
. 
is called a regular amicable pair of type 
if
 |
(13)
|
where 
is the greatest common divisor,
 |
(14)
|
and 
are squarefree, then the number
of prime factors of 
and 
are 
and 
. Pairs which are not regular are called irregular or exotic
(te Riele 1986). There are no regular pairs of type 
for 
. If 
and
 |
(15)
|
is even, then 
cannot be an amicable pair (Lee 1969). The minimal and
maximal values of 
found by te Riele (1986) were
 |
(16)
|
and
 |
(17)
|
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
 |
(18)
|
te Riele (1986) found no amicable 
-tuples having the same pair sum
for 
.
However, Moews and Moews found a triple in 1993, and te Riele found a quadruple in
1995. In November 1997, a quintuple and sextuple were discovered. The sextuple is
(1953433861918, 2216492794082), (1968039941816, 2201886714184), (1981957651366, 2187969004634),
(1993501042130, 2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962),
all having pair sum 4169926656000. Amazingly, the sextuple
is smaller than any known quadruple or quintuple, and is likely smaller than any
quintuple.
The earliest known odd amicable numbers all were divisible by 3. This led Bratley and McKay (1968) to conjecture that there are no amicable pairs coprime to 6 (Guy 1994, p. 56). However, Battiato and Borho (1988) found a counterexample, and now many amicable pairs are known which are not divisible by 6 (Pedersen). The smallest known example of this kind is the amicable pair (42262694537514864075544955198125, 42405817271188606697466971841875), each number of which has 32 digits.
A search was then begun for amicable pairs coprime to 30. The first example was found by Y. Kohmoto in 1997, consisting of a pair of numbers each having 193 digits
(Pedersen). Kohmoto subsequently found two other examples, and te Riele and Pedersen
used two of Kohmoto's examples to calculated 243 type-
pairs coprime to 30 by means of a method which generates
type-
pairs from a type-
pairs.
No amicable pairs which are coprime to 
are currently known.
The following table summarizes the largest known amicable pairs discovered in recent years. The largest of these is obtained by defining
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(19)
|
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(20)
|
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(21)
|
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(22)
|
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(23)
|
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(24)
|
then 
,
, 
and 
are all primes, and the numbers
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(25)
|
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(26)
|
are an amicable pair, with each member having 
decimal digits (Jobling 2005).
| digits | date | reference |
| 4829 | Oct. 4, 1997 | M. García |
| 8684 | Jun. 6, 2003 | Jobling and Walker 2003 |
| 16563 | May 12, 2004 | Walker et al. 2004 |
| 17326 | May 12, 2004 | Walker et al. 2004 |
| 24073 | Mar. 10, 2005 | Jobling 2005 |
Amicable pairs in Gaussian integers also exist, for example
 |  |  |
(27)
|
 |  |  |
(28)
|
and
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(29)
|
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(30)
|
(T. D. Noe, pers. comm.).
."
Math. Comput. 56, 859-865, 1991.Dickson, L. E. 
." Rev. Jan. 8, 1993b. 
." Math. Comput. 47,
361-368 and S9-S35, 1986.te Riele, H. J. J.; Borho, W.; Battiato,
S.; Hoffmann, H.; and Lee, E. J. "Table of Amicable Pairs Between 
and 
." Centrum voor Wiskunde en Informatica, Note NM-N8603.
Amsterdam: Stichting Math. Centrum, 1986.te Riele, H. J. J.
"A New Method for Finding Amicable Pairs." In