| 释义 |
Amicable PairAn 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 pairsare occasionally called Friendly Pairs, although this nomenclature is to be discouraged sinceFriendly Pairs are defined by a different, if related, criterion. Symbolically, amicable pairssatisfy
where  is the Restricted Divisor Function or, equivalently,
 | (3) |
 is the Divisor Function. The smallest amicable pair is (220, 284) which has factorizations
giving Restricted Divisor Functions
The quantity
 | (8) |
 , is called the Pair Sum.
In 1636, Fermat  found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056). By1747, 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 A002025and A002046). An exhaustive tabulation is maintainedby D. Moews.
Let an amicable pair be denoted  with  . is called a regular amicable pair of type  if
 | (9) |
 is the Greatest Common Divisor,
 | (10) |
 and  are Squarefree, then the number of Prime factors of and are  and  . Pairs which arenot regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type  for . If  and
 | (11) |
 found by te Riele(1986) were
 | (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) |
 -tuples having the same Pair Sum for  . However, Moews and Moews found a triplein 1993, and te Riele found a quadruple in 1995. In November 1997, a quintuple and sextuple were discovered. The sextupleis(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, andis likely smaller than any quintuple.
On October 4, 1997, Mariano Garcia found the largest known amicable pair, each of whose members has 4829Digits. The new pair is
where
 ,  ,  , and  are Prime.
Pomerance (1981) has proved that
 | (21) |
References
Alanen, J.; Ore, Ø.; and Stemple, J. ``Systematic Computations on Amicable Numbers.'' Math. Comput. 21, 242-245, 1967.Battiato, S. and Borho, W. ``Are there Odd Amicable Numbers not Divisible by Three?'' Math. Comput. 50, 633-637, 1988. Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 62 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. 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. 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 (,1).'' Math. Comput. 56, 859-865, 1991. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 38-50, 1952. Erdös, P. ``On Amicable Numbers.'' Publ. Math. Debrecen 4, 108-111, 1955-1956. Erdös, P. ``On Asymptotic Properties of Aliquot Sequences.'' Math. Comput. 30, 641-645, 1976. 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. Lee, E. J. ``Amicable Numbers and the Bilinear Diophantine Equation.'' 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. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 145 and 155-156, 1979. Moews, D. and Moews, P. C. ``A Search for Aliquot Cycles and Amicable Pairs.'' Math. Comput. 61, 935-938, 1993. Moews, D. and Moews, P. C. ``A List of Amicable Pairs Below .'' Rev. Jan. 8, 1993. http://xraysgi.ims.uconn.edu:8080/amicable.txt. Moews, D. and Moews, P. C. ``A List of the First 5001 Amicable Pairs.'' Rev. Jan. 7, 1996. http://xraysgi.ims.uconn.edu:8080/amicable2.txt. Ore, Ø. Number Theory and Its History. New York: Dover, pp. 96-100, 1988. Pedersen, J. M. ``Known Amicable Pairs.'' http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm. Pomerance, C. ``On the Distribution of Amicable Numbers.'' J. reine angew. Math. 293/294, 217-222, 1977. Pomerance, C. ``On the Distribution of Amicable Numbers, II.'' J. reine angew. Math. 325, 182-188, 1981. Scott, E. B. E. ``Amicable Numbers.'' Scripta Math. 12, 61-72, 1946. Sloane, N. J. A. SequencesA002025/M5414and A002046/M5435in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. te Riele, H. J. J. ``On Generating New Amicable Pairs from Given Amicable Pairs.'' Math. Comput. 42, 219-223, 1984. te Riele, H. J. J. ``Computation of All the Amicable Pairs Below .'' 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 Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (Vancouver, BC, August 9-13, 1993) (Ed. W. Gautschi). Providence, RI: Amer. Math. Soc., pp. 577-581, 1994.  Weisstein, E. W. ``Sociable and Amicable Numbers.'' Mathematica notebook Sociable.m.
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