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Topic: Perfect number

	Perfect number - Wikipedia, the free encyclopedia
		In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number.
		Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n.
		By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence.
	en.wikipedia.org /wiki/Perfect_number (1176 words)

	NationMaster - Encyclopedia: Perfect number
		Thus, 6 is a perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6.
		Numbers where the sum is less than the number itself are called deficient, and where it is greater, abundant.
		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/Perfect-number (2388 words)

	Multiply perfect number - Wikipedia, the free encyclopedia
		In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
		For a given natural number k, a number n is called k-perfect (or k-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect.
		This implies that if an integer n is a 3-perfect number divisible by 2 but not by 4, then n/2 is an odd perfect number, of which none are known.
	en.wikipedia.org /wiki/Multiply_perfect_number (218 words)

	Perfect number (Site not responding. Last check: 2007-10-22)
		In mathematics, a perfect number is an integer which is the sumof its proper positive divisors, excluding itself.
		Thus, 6 is a perfect number, because 1, 2 and 3 are its proper positive divisors and1 + 2 + 3 = 6.
		Numbers where the sum is less than the numberitself are called deficient, and where it is greater, abundant.
	www.therfcc.org /perfect-number-34803.html (512 words)

	perfect number
		As René Descartes pointed out: "Perfect numbers like perfect men are very rare." All end in 6 or 8, though what seems to be an alternating pattern of 6's and 8's for the first few perfect numbers doesn't continue.
		An irreducible semi-perfect number is a semi-perfect number, none of whose factors is semi-perfect, e.g.
		A multiply perfect number is a number n whose divisors sum to a multiple of n.
	www.daviddarling.info /encyclopedia/P/perfect_number.html (405 words)

	Perfect number
		A perfect number is a number which is the sum of its proper divisors.
		Perfect numbers are related to Mersenne primes (prime numbers that are one less than a power of 2): if M is a Mersenne prime, then M×(M+1)/2 is a perfect number.
		Numbers where the sum is less than the number itself are called deficient, and where it is greater, abundant; these, together with perfect numbers, come from Greek numerology.
	www.fastload.org /pe/Perfect_number.html (375 words)

	Perfect number (Site not responding. Last check: 2007-10-22)
		In mathematics a perfect number is an integer which is the sum of its positive divisors excluding itself.
		Thus 6 is a perfect number because 1 and 3 are its proper positive divisors 1 + 2 + 3 = 6.
		Numbers where the is less than the number itself are deficient and where it is greater abundant.
	www.freeglossary.com /Perfect_number (589 words)

	Perfect numbers
		It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity.
		Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.
		The four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Perfect_numbers.html (4360 words)

	Straight Dope Staff Report: What's the story on perfect numbers?
		Perfect numbers are a holdover from the days of the Pythagoreans, when mathematicians were mystics as much as anything else and put a lot more stock in coincidence.
		Quasiperfect numbers have a divisor sum of 2n+1; almost perfect numbers have a divisor sum of 2n-1.
		For example, sublime numbers have a perfect number of divisors, and the sum of their divisors is itself perfect.
	www.straightdope.com /mailbag/mperfectnumbers.html (683 words)

	Perfect Number Analyzer (Site not responding. Last check: 2007-10-22)
		An abundant number is a positive integer for which the sum of all its proper divisors (factors) is greater than the number.
		A deficient number is a positive integer for which the sum of all its proper divisors (factors) is less than the number.
		A perfect number is a positive integer for which the sum of all its proper divisors (factors) equals the number.
	britton.disted.camosun.bc.ca /perfect/jbperfect.htm (167 words)

	Perfect Number -- from MathWorld
		Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid.
		Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes, which are prime numbers of the form
		It is known that all even perfect numbers (except 6) end in 16, 28, 36, 56, 76, or 96 (Lucas 1891) and have digital root 1.
	users.skynet.be /fa956617/math/topics/PerfectNumber.html (710 words)

	Multiply perfect number (Site not responding. Last check: 2007-10-22)
		For a given natural number k a number n is called k -perfect (or k -fold perfect) iff the sum of all positive divisors of n (the divisor function σ(n)) is equal to kn ; a number is thus perfect iff it is 2-perfect.
		A number that k -perfect for a certain k is called a multiply perfect number.
		This implies that if an integer n is a 3-perfect number divisible by but not by 4 then n /2 is an odd perfect number of which none are known.
	www.freeglossary.com /Multiply_perfect_number (459 words)

	Perfect Number (Site not responding. Last check: 2007-10-22)
		A Perfect number is a number which has all its integer factors added together to result in the source number.
		For example to test if a number is perfect all of its integer factors need to be found then added together, not including the original number, and if the result matches the original number then the number is perfect.
		The other neat thing about Perfect number search algorithms is that they keep a CPU busy for an extended length of time which makes it great for testing and bench-marking new computer systems.
	www.cate.com.au /download/perfect.html (358 words)

	Math Forum: Ask Dr. Math FAQ: Perfect Numbers
		We do know that there are an infinite number of prime numbers, which means there is a very high chance that there are an infinite number of perfect numbers.
		A Mersenne Number is a number that is equal to one less than a power of 2, or (2^n-1) where n is any positive integer.
		The rest of the factors of the perfect number 2^(n-1) * (2^n -1) are each of the above factors multiplied by the prime 2^(n-1).
	www.mathforum.org /dr.math/faq/faq.perfect.html (2022 words)

	PlanetMath: perfect number
		It is not known if there are any odd perfect numbers, but all even perfect numbers have been classified as follows:
		is perfect, and every even perfect number is of this form.
		This is version 10 of perfect number, born on 2001-10-15, modified 2006-08-16.
	www.planetmath.org /encyclopedia/PerfectNumber.html (98 words)

	History Of Perfect Numbers
		Perfect Numbers - the sum of the numbers aliquot parts is equal to the number.
		The sixth Perfect Number was discovered in 1555 by J Scheyble, this however, was not noticed untill 1977 so therefore had no influence on the progress of the Perfect Numbers research.
		Even thought this is the 39th Perfect Number to be discovered it may not be the 39th Perfect Number as not all smaller cases have been ruled out.
	www.bath.ac.uk /~ma2le/History.html (1527 words)

	The Prime Glossary: perfect number
		One example is the perfect numbers, those integers which are the sum of their positive proper divisors.
		Whatever significance ascribed to them, these three perfect numbers above, and 8128, were known to be "perfect" by the ancient Greeks, and the search for perfect numbers was behind some of the greatest discoveries in number theory.
		While seeking perfect and amicable numbers, Pierre de Fermat discovered Fermat’s Little Theorem, and communicated a simplified version of it to Mersenne in 1640.
	primes.utm.edu /glossary/page.php?sort=PerfectNumber (318 words)

	Perfect numbers
		Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
		This showed that Nicomachus's first assertion is false since the fifth perfect number has 8 digits.
		The search for perfect numbers had now become an attempt to check whether Mersenne was right with his claims in Cogitata physica mathematica.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Perfect_numbers.html (4360 words)

	Ivars Peterson's MathLand
		Among those of special interest were the perfect numbers, which have the property that their proper divisors add up to the number itself.
		For example, the final digits of the four perfect numbers that he knew alternate between 6 and 8.
		Lists of known perfect numbers are available at http://forum.swarthmore.edu/dr.math/problems/perfect.html and http://www.maths.uts.edu.au/number/perfect.html.
	www.maa.org /mathland/mathland_1_27.html (781 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		The concepts even, odd, Mersenne prime, perfect numbers were extended as follows: (i) $\eta$ is an even Gaussian integer if $(1+i)\eta$ and an odd integer if $(1+i)\nmid\eta$.
		If a (norm-) perfect number $\eta$ does not have a (norm-) perfect number as a proper divisor, then $\eta$ is primitive.
		Corollary: $\eta$ is a primitive perfect number if and only if there exists a rational prime $p\equiv 1 (\text{mod}\,8)$ such that $\eta=(1+i)^{p-1}M_p$.
	www.mathpuzzle.com /GaussianPerfect.txt (281 words)

	Perfect Number
		A "perfect" number is a number whose divisors sum to twice itself.
		Euclid was one of the first to study perfect numbers, although, due to the difficulty of finding such numbers, he only knew of the first four perfect numbers: 6, 28, 496, and 8128.
		In his book Perfect Numbers, Richard Shoemaker defines social numbers "as being a chain of numbers in which each number is equal to the sum of all the proper divisors of the preceding number, the last being considered as preceding the first number of the chain" (page 27).
	math.arizona.edu /~ura/001/gaberdiel.jw (6796 words)

	Multiply Perfect Numbers
		Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n.
		And a further list of multiply perfect numbers sorted only by their factorizations (built from the 5189 MPNs of the master list and gziped 130 kB).
		To verify these numbers, three steps must be taken: 1) verify for each number n given in its prime factorization that all factors are really prime numbers, 2) compute o(n) which envolves the factorization of large numbers and then check n's claimed abundancy 3) and lastly test whether n is really new.
	www.uni-bielefeld.de /~achim/mpn.html (1723 words)

	id:A000396 - OEIS Search Results
		Perfect numbers: equal to sum of proper divisors.
		Thus perfect numbers N, being M-th triangular, have form (6t+1)(3*t+1), whence the property N (mod 9)=1 for all N after the first.
		The numbers 2^(p-1)(2^p - 1) are perfect, where p is a prime such that 2^p - 1 is also prime (see A000043, the Mersenne primes).
	www.research.att.com /~njas/sequences/A000396 (692 words)

	Puzzle 111. Spoof odd Perfect numbers
		The method for finding such numbers is similar to the method for finding multiperfect numbers: Write down a small composite number (spoof prime factor) p and above it write down p+1.
		I have found number of solutions(29 which are less than 410^11) with 7 composite factors like 172368000=46891475*95.
		In return to my invitation Shyam asks to look for odd and completely spoof perfect numbers or at least bounds below which such a number can not exist; but he believes that there are none.
	www.primepuzzles.net /puzzles/puzz_111.htm (575 words)

	Numerical Geometry
		Triangular Number is the sum of all the numbers from 1 to n.
		Square Numbers are also associated the polar concepts of Confinement (due to the limitations imposed by structure) and Expansion (in the sense of the Four Directions).
		This Number is the basis of the Unity Holograph which is built upon four nested word clusters that sum to multiples of the Number (= One = Love).
	www.biblewheel.com /GR/GR_Figurate.asp (1576 words)

	SS > factoids > perfect number
		A perfect number P is equal to the sum of its divisors (where the divisors include 1, but not P itself).
		Every even perfect number ends in a '6' or an '8'.
		Every even perfect number, other than 6, is the sum of consecutive odd cubes.
	public.logica.com /~stepneys/cyc/p/perfect.htm (364 words)

	Mathematics Enrichment Workshop: The Perfect Number Journey (Site not responding. Last check: 2007-10-22)
		A perfect number is then obtained by multiplying this sum to the last power of 2.
		In the exercise that follows, you are going to use this method to determine the next two perfect numbers.
		Before you proceed to find the fifth perfect number, you may want to pause for a moment and take a closer look at the first four perfect numbers that have been obtained this way.
	home1.pacific.net.sg /%7Enovelway/MEW2/lesson1.html (522 words)

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