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Topic: Amicable number

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	Amicable number (Site not responding. Last check: 2007-11-05)
		Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equalto the other, unity being considered as a proper divisor but not the number itself.
		Such a pair is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which thesum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220.
		Amicable numbers have been studied by Al Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980 - 1037), RenéDescartes (1596 - 1650), to whom the formula ofThabit is sometimes ascribed, C. Rudolphus and others.
	www.therfcc.org /amicable-number-218008.html (233 words)

	Amicable number - Wikipedia, the free encyclopedia
		Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself.
		In every known case, the numbers of a pair are either both even or both odd, though there is no known reason why an even-odd pair could not exist.
		Amicable numbers have been studied by Al Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980-1037), René Descartes (1596-1650), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others.
	en.wikipedia.org /wiki/Amicable_numbers (345 words)

	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7257 words)

	Amicable number (Site not responding. Last check: 2007-11-05)
		Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to other unity being considered as a proper but not the number itself.
		Such a is (220 284); for the proper divisors of 220 1 2 4 5 10 11 20 44 55 and 110 of which the is 284; and the proper divisors of are 1 2 4 71 and 142 which the sum is 220.
		Amicable numbers have been studied by Al (died 1007) Abu Mansur Tahir al-Baghdadi (980 - 1037) René Descartes (1596 - 1650) to whom the formula of Thabit sometimes ascribed C. Rudolphus and others.
	www.freeglossary.com /Amicable_number (548 words)

	Amicable number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Amicable numbers are two (A concept of quantity derived from zero and units) numbers so related that the (A quantity of money) sum of the proper (The number by which a dividend is divided) divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself.
		Amicable numbers were known to the (Click link for more info and facts about Pythagoreans) Pythagoreans, who accredited them with many mystical properties.
		If a number equals the sum of its own proper divisors, it is called a (Click link for more info and facts about perfect number) perfect number.
	www.absoluteastronomy.com /encyclopedia/a/am/amicable_number.htm (194 words)

	Encyclopedia: Amicable-number
		The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3,...) and the number zero.
		In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself.
		In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself.
	www.nationmaster.com /encyclopedia/Amicable_number (629 words)

	Amicable Numbers by Shyam Sunder Gupta
		The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras.
		Two numbers are called Amicable (or friendly) if each equals to the sum of the aliquot divisors of the other (aliquot divisors means all the divisors excluding the number itself).
		Harshad (or Niven) numbers are those numbers which are divisible by their sum of the digits.
	www.shyamsundergupta.com /amicable.htm (1118 words)

	Untitled Document (Site not responding. Last check: 2007-11-05)
		The purpose of this activity is to emphasize skills of factoring numbers, relating numbers to one another, and to serve as an interesting application of previously learned knowledge.
		Number theory is a term used to denote the abstract side of number study (Eves).
		Recall that the proper divisor is all divisors of a number N, except for N itself.
	jwilson.coe.uga.edu /EMAT4500/AssignmentPages/MaryBethWiggins/starter1.html (493 words)

	[No title]
		Amicable numbers are 2-periodic orbits of the sigma function sigma(n) which is the sum of the divisors of n.
		The Carmichael numbers under 100'000 are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361.
		Perfect numbers are fixed points, amicable numbers are periodic orbits.
	www.math.harvard.edu /~knill/sofia/data/number.txt (1516 words)

	Ivars Peterson's MathTrek -Appealing Numbers
		The next perfect number is 28, which is the sum of 1, 2, 4, 7, and 14.
		The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284; and 284 is evenly divisible by 1, 2, 4, 71, and 142, which add up to 220.The Pythagorean brotherhood regarded 220 and 284 as numerical symbols of friendship.
		The pair, consisting of the numbers 1,184 and 1,210, was discovered by a 16-year-old Italian youth, probably using trial and error.
	www.maa.org /mathland/mathtrek_2_26_01.html (626 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-05)
		Date: 11/12/98 at 20:05:58 From: Doctor White Subject: Re: Amicable numbers Amanda: Amicable numbers are a pair of numbers such that the sum of their proper divisors (not including itself) equals the other number.
		Because amicable numbers must come in pairs, they are sometimes called amicable pairs or amicable partners.
		Thus, 220 and 284 are a pair of amicable numbers.
	mathforum.org /library/drmath/view/52560.html (361 words)

	Al-Farisi (Site not responding. Last check: 2007-11-05)
		On the other hand his contributions to number theory are discussed the references [2], [3], [4], [7], [8], and [9], most of which are also written by R Rashed but relate to discoveries made after the article [1] was written.
		At the end of his treatise al-Farisi gives the pairs of amicable numbers 220, 284 and 17296, 18416, obtained from using Thabit's rule with n = 2 and n = 4 respectively.
		The pair of amicable number 17296, 18416 are known as Euler's amicable pair.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Al-Farisi.html (1027 words)

	The Prime Glossary: amicable numbers
		They call certain numbers amicable numbers, adopting virtues and social qualities to numbers, such as 284 and 220; for the parts of each have the power to generate the other.
		Pythagoras is reported to have said that a friend is "one who is the other I, such as are 220 and 284." Now amicable numbers are most often (and most properly!) relegated to the exercise sections of elementary number theory texts.
		In 1909 one of his pairs was found to be not amicable, and in 1914 the same fate took a second pair.
	primes.utm.edu /glossary/page.php?sort=AmicableNumber (427 words)

	Math words page 13
		Aliquot parts of a number are proper divisors of the number that are smaller than the number.
		The numbers were inscribed on "magic charms" in the middle ages which were sold to insure the fidelity of ones lover.
		Number systems are also describe by the number of characters in the system.
	www.pballew.net /arithm13.html (4167 words)

	Amicable number - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-11-05)
		are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself.
		Amicable numbers have been studied by Al_Madshritti (died 1007), Abu Mansur Tahir al-Baghdadi (980-1037), René_Descartes (1596-1650), to whom the formula of Thabit is sometimes ascribed, C. Rudolphus and others.
		If a number equals the sum of ''its own'' proper divisors, it is called a perfect_number.
	www.indexsuche.com /Amicable_number.html (217 words)

	1000 (number) - Open Encyclopedia (Site not responding. Last check: 2007-11-05)
		1681 - 41^2, smallest number yielded by the formula n^2 + n + 41 that is not a prime
		In the decimal expansion of $e, the first time all 10 digits appear in sequence starts at the 1729th decimal place.$
		1791 - largest natural number that cannot be expressed as a sum of at most four hexagonal numbers.
	open-encyclopedia.com /One_thousand (625 words)

	Read about Category:Number theory at WorldVillage Encyclopedia. Research Category:Number theory and learn about ... (Site not responding. Last check: 2007-11-05)
		number theory is that branch of pure mathematics concerned with the properties of
		More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of integers.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
	encyclopedia.worldvillage.com /s/b/Category:Number_theory (118 words)

	Aliquot Sums (Site not responding. Last check: 2007-11-05)
		The aliquot divisors of a number are all of its divisors except the number itself.
		A number whose aliquot sum equals its value is a PERFECT number (6 for example).
		Amicable pairs are fairly common but search times get long when we get up in the millions.
	www.delphiforfun.org /Programs/AliquotSums.htm (732 words)

	Perfect, amicable and sociable numbers
		For a number n, we define s(n) to be the sum of the aliquot parts of n, i.e., the sum of the positive divisors of n, excluding n itself: so, for example, s(8)=1+2+4=7, and s(12)=1+2+3+4+6=16.
		An amicable pair is a cycle of length 2 of s, i.e., a pair of numbers each of which equals the sum of the other's aliquot parts; the members of amicable pairs are also called amicable.
		However the sum of unitary divisors, bi-unitary divisors, or infinitary divisors of a number is always even, unless the number is a power of 2.
	djm.cc /amicable.html (2519 words)

	F_rcase.html (Site not responding. Last check: 2007-11-05)
		The ancient Greeks believed that amicable numbers had a special influence on establishing friendships between individuals.
		The name amicable may have originated from the Latin word for friend which is amici, since educated men wrote and read in Latin at that period of time.
		Since computers have taken over, it is known that there are 42 amicable pairs smaller than a million, 108 pairs smaller than 10 million, and 236 pairs smaller than 100 million.
	math.ucsd.edu /~jwavrik/proj107b/F_rcase4.html (143 words)

	Amicable - Definition of Amicable by Webster's Online Dictionary (Site not responding. Last check: 2007-11-05)
		the amicable manner in which it was managed.
		(Law.) an action commenced and prosecuted by amicable consent of the parties, for the purpose of obtaining a decision of the court on some matter of law involved in it.
		(Math.) two numbers, each of which is equal to the sum of all the aliquot parts of the other.
	www.webster-dictionary.org /definition/amicable (127 words)

	Computational number theory
		The six largest numbers factored so far with NFS have 174 (factored on December 3, 2003), 160 (factored on April 1, 2003), 158 (factored on January 19, 2002), 155, 140, and 130 decimal digits.
		The previous three SNFS records (established on November 14, 2000, on April 8, 1999 and in September 1998) are numbers of 233, 211, and 186 decimal digits, respectively.
		The number and the size of known amicable number pairs has grown explosively in recent years: from 1108 amicable pairs in 1972 (the largest pair consisting of two 25-digit numbers), to more than five million amicable pairs in May 2003 (the largest pair consisting of two 5577-digit numbers).
	db.cwi.nl /projecten/project.php4?prjnr=84 (1003 words)

	Ancient Greeks: Prime Numbers and Number Theory
		The Pythagoreans saw the number one as the primordial unity from which all else is created.
		Diophantus noticed that numbers of the form 4n + 3 cannot be the sum of two squares and 24n + 7 cannot be the sum of three squares.
		Plato's Geometrical Number, The Tyrant and the Bride
	www.mlahanas.de /Greeks/Primes.htm (2678 words)

	Definition of 1 E4
		Ten thousand (10000) is the natural number following 9999 and preceding 10001.
		The Greek prefix for 10,000 is myria-, from which myriad is derived.
		78557 - conjectured to be the smallest Sierpinski number
	www.wordiq.com /definition/1_E4 (455 words)

	Tables of Aliquot Cycles
		Note 1: All powers of 2 are augmented perfect numbers; no other augmented perfect numbers are known.
		Note 2: Multiplying all members of an exponential aliquot cycle by a squarefree number prime to all the members of the cycle yields a new exponential aliquot cycle.
		For amicable numbers all pairs with smaller member below the limit are known.
	amicable.homepage.dk /tables.htm (200 words)

	Various AP Lists and Statistics (Site not responding. Last check: 2007-11-05)
		Number of amicable pairs of each type with smallest member below 10
		Pairs shown in italics are not necesarily the smallest pair of the type.
		An amicable pair (M,N) = (gm,gn) pair with g = gcd(M,N) is said to have type (i,j) if m has i different prime factors not dividing g, and n has j different prime factors not dividing g.
	amicable.homepage.dk /apstat.htm (407 words)

	ipedia.com: Deficient number Article (Site not responding. Last check: 2007-11-05)
		In mathematics, a deficient number or defective number is a number n for which σ n.
		In mathematics, a deficient number or defective number is a number n for which σ(n)
		For example, all prime numbers, all prime powers and all proper divisors of deficient or perfect numbers are deficient.
	www.ipedia.com /deficient_number.html (143 words)

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