An amicable pair (m, n) consists of two integers m, n 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 (m, n) 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) |
The quantity
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(9) |
in this case, 220 + 284 = 504, 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), ... (Sloane's 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 1952, p. 47). There were 390 known amicable pairs as of 1946 (Escott 1946). There are a total of 236 amicable pairs below 108 (Cohen 1970), 1427 below 1010 (te Riele 1986), 3340 less than 1011 (Moews and Moews 1993), 4316 less than 
(Moews and Moews), and 5001 less than 
(Moews and Moews).
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
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(10) |
for large enough n (Guy 1994). No nonfinite lower bound has been proven.
Let an amicable pair be denoted (m, n), and take m < n. (m, n) is called a regular amicable pair of type (i, j) if
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(11) |
where 
is the greatest common divisor,
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(12) |
M and N are squarefree, then the number of prime factors of M and N are i and j. Pairs which are not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type 
for 
.
and
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(13) |
is even, then (m, n) cannot be an amicable pair (Lee 1969). The minimal and maximal values of 
found by te Riele (1986) were
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(14) |
and
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(15) |
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
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(16) |
te Riele (1986) found no amicable n-tuples having the same pair sum for n > 2. 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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(17) |
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(18) |
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(19) |
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(20) |
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(21) |
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(22) |
then p, q,
and 
are all primes, and the numbers
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(23) |
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(24) |
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 |
Aliquot Sequence, Amicable Quadruple, Amicable Triple, Augmented Amicable Pair, Breeder, Crowd, Euler's Rule, Friendly Pair, Multiamicable Numbers, Pair Sum, Quasiamicable Pair, Rational Amicable Pair, Sociable Numbers, Super Unitary Amicable Pair, Thâbit ibn Kurrah Rule, Unitary Amicable Pair
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Borho, W. "On Thabit ibn Kurrah's Formula for Amicable Numbers." Math. Comput. 26, 571-578, 1972.
Borho, W. "Some Large Primes and Amicable Numbers." Math. Comput. 36, 303-304, 1981.
Borho, W. "Befreundete Zahlen: Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie." In Mathematische Miniaturen 1: Lebendige Zahlen: Fünf Exkursionen. Basel, Switzerland, Birkhäuser, pp. 5-38, 1981.
Borho, W. and Hoffmann, H. "Breeding Amicable Numbers in Abundance." Math. Comput. 46, 281-293, 1986.
Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers and Their Distribution." Math. Comput. 24, 431-432, 1970.
Bratley, P. and McKay, J. "More Amicable Numbers." Math. Comput. 22, 677-678, 1968.
Cohen, H. "On Amicable and Sociable Numbers." Math. Comput. 24, 423-429, 1970.
Costello, P. "Amicable Pairs of Euler's First Form." J. Rec. Math. 10, 183-189, 1977-1978.
Costello, P. "Amicable Pairs of the Form 
.
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 38-50, 1952.
Dubner, H. "New Amicable Pair Record." NMBRTHRY@listserv.nodak.edu posting. Oct. 14, 1997. http://listserv.nodak.edu/scripts/wa.exe?A2=ind9710&L=NMBRTHRY&F=&S=&P=695.
Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4, 108-111, 1955-1956.
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Escott, E. B. E. "Amicable Numbers." Scripta Math. 12, 61-72, 1946.
García, M. "New Amicable Pairs." Scripta Math. 23, 167-171, 1957.
Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 160-171, 1978.
Guy, R. K. "Amicable Numbers." §B4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 55-59, 1994.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998.
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Jobling, P. "A New Largest Known Amicable Pair." http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0503&L=nmbrthry&F=&S=&P=552.
Lee, E. J. "Amicable Numbers and the Bilinear Diophantine Equation." NMBRTHRY@listserv.nodak.edu posting. Mar. 10, 2005. Math. Comput. 22, 181-197, 1968.
Lee, E. J. "On Divisibility of the Sums of Even Amicable Pairs." Math. Comput. 23, 545-548, 1969.
Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, I." J. Rec. Math. 5, 77-93, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, II." J. Rec. Math. 5, 153-173, 1972.
Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, III." J. Rec. Math. 5, 231-249, 1972.
Lee, E. J. and Madachy, J. S. Errata to "The History and Discovery of Amicable Numbers, I-III." J. Rec. Math. 6, 53, 164, and 229, 1973.
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 145 and 155-156, 1979.
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Root, S. Item 61 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item61.
Sloane, N. J. A. Sequences A002025/M5414 and A002046/M5435 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Souissi, M. Un Texte Manuscrit d'Ibn Al-Banna' Al-Marrakusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975.
Speciner, M. Item 62 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 24, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item62.
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