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	perfect number
		A whole number that is equal to the sum of all its factors except itself.
		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.
		A multiply perfect number is a number n whose divisors sum to a multiple of n.
	www.daviddarling.info /encyclopedia/P/perfect_number.html (361 words)

	Carl Pomerance
		On the number of false witnesses for a composite number, P.
		On the role of smooth numbers in number theoretic algorithms, C.
		Smooth numbers and the quadratic sieve, C. Pomerance, to appear in the proceedings of an MSRI workshop, J. Buhler and P. Stevenhagen, eds.
	www.math.dartmouth.edu /~carlp (2315 words)

	Multiply Perfect Numbers
		Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n.
		If o(n) is an integral multiply of n, then n is denoted as a multiply perfect number or k-fold perfect number (also called multiperfect number or pluperfect number).
		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 (1714 words)

	Decimals, Whole Numbers, and Exponents
		In the number 3.762, the 3 is in the ones place, the 7 is in the tenths place, the 6 is in the hundredths place, and the 2 is in the thousandths place.
		The numbers 1, 4, 9, 16, and 25 are all perfect squares.
		The numbers 1, 8, 27, 64, and 125 are all perfect cubes.
	www.mathleague.com /help/decwholeexp/decwholeexp.htm (1681 words)

	Perfect number Summary
		Perfect number-a number that is the sum of its proper divisors (a proper divisor is a divisor smaller than the number itself).
		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.
	www.bookrags.com /Perfect_number (1658 words)

	smallest perfect number : Search Results : GoodSearch : Search the Internet with GoodSearch and support your favorite ...
		The smallest perfect number is 6, which is the sum of 1, 2, and 3.
		are an infinite number of exponential perfect numbers, of exponential amicable...
		Recall that a perfect number is an integer that is the sum of its aliquot...
	www.goodsearch.com /Search.aspx?Keywords=smallest%20perfect%20number (281 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.
		A number that is k-perfect for a certain k is called a multiply perfect number.
	en.wikipedia.org /wiki/Multiply_perfect_number (218 words)

	1 (number) - Biocrawler (Site not responding. Last check: )
		1 (one) is a number, numeral, and glyph.
		One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few.
		It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences.
	www.biocrawler.com /encyclopedia/One (1728 words)

	Perfect numbers
		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.
		Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect.
		However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid's time and continuing till Nicomachus wrote his treatise.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Perfect_numbers.html (4360 words)

	math lessons - Multiply perfect number
		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.
		For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-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.
	www.mathdaily.com /lessons/Multiply_perfect_number (198 words)

	Mathematics Archives - Numbers
		The 47 Society is an international interest-group that follows the occurence and recurrence of the quintessential random number: 47.
		In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse.
		A prime k-tuplet is a sequence of consecutive prime numbers {p1, p2,..., pk} such that, in some sense, pk - p1 is as small as possible.
	archives.math.utk.edu /subjects/numbers.html (1310 words)

	Perfect Numbers :: Factoring : Gourt
		Six (6) is the first perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6.
		Odd Perfect Numbers - Missing Values - Status of various factorisations that would extend the known area of non-existence of odd perfect numbers, by Kevin Hare.
		Perfect Numbers - A detailed history of the quest for perfect numbers, from Euclid to their present-day Mersenne discoveries.
	science.gourt.com /Math/Number-Theory/Factoring/Perfect-Numbers.html (411 words)

	[No title]
		Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
		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.
		A perfect number is defined to be one which is equal to the sum of its aliquot parts.
	www.resonancepub.com /perfectnums.htm (4291 words)

	Notable Properties of Specific Numbers at MROB
		This number is equal to the sum of the 10th powers of each of its digits, and is unique in being the only 10-digit number to meet this requirement.
		These numbers are really hard to estimate because astronomers cannot see most of the stars in our galaxy (due to clouds of dust, and because most stars are too faint) or most of the galaxies (same reasons) so it ends up being the product of a lot of statistical guess-work.
		This is the number of carbon atoms in 12 grams of pure carbon, or the number of atoms in N grams of an element with atomic weight N.
	home.earthlink.net /~mrob/pub/numbers-6.html (2805 words)

	Integer Bars: More About Multiplication
		Perfect Squares - Mathematically, a perfect square is when you multiply a two numbers that are the same.
		To use the integer bars to find a perfect square, you can follow the methods described in Activity 1 and you will end up with an image that is a perfect square.
		The perfect square on the left is made of 4 bars of size 4.
	www.arcytech.org /java/integers/multiplication2.html (572 words)

	Puzzle 332. Odd abundance.
		To find an odd n such that, for sigma(n)/n = 2, we would have to find an odd perfect number (of which none are known to exist).
		An old and unproved conjecture states that the only odd multiply perfect number is 1 - which extends the ancient Greek's conjecture that states the only odd perfect number is 1.
		Conversely this is not true since an hypothetical odd multiperfect number may not lead to a solution!.
	www.primepuzzles.net /puzzles/puzz_332.htm (485 words)

	Arithmetic, Numeration, Number Theory - Numericana
		Since the number 9N divides the number which consists of P nines followed by a certain number J of zeroes, N divides the number consisting of P ones followed by J zeroes, and also the integer composed of P sevens followed by J zeroes.
		The next two numbers in the list, the 13th and 14th Mersenne primes, are much larger (corresponding to n=521 and n=607) and were both discovered the same day (January 30, 1952, around 22:00 PST and shortly before midnight) by Raphael Mitchel Robinson (1911-1995), at the dawn of the computer age.
		This coefficient is indeed obtained by counting the number of ways there is to choose an exponent multiple of 1 from the first factor, a multiple of 2 from the second factor, a multiple of 3 from the third, etc. so these exponents add up to n.
	home.att.net /~numericana/answer/numbers.htm (7644 words)

	Finding Perfect Numbers (Site not responding. Last check: )
		If the sum of the factors is less than the number itself, then this number is considered "deficient." If the sum of the factors is more than the number itself, then this number is considered "abundant." If the sum of the factors equals the number, then this number is considered a "perfect" number.
		This lesson on deficient, abundant, and perfect numbers is often optional in the fifth through eighth grade curriculum.
		For example, a number is odd or even, composite or prime, exponential, squared, cubed, etc. Students also will be able to note a pattern for perfect numbers.
	www.wpunj.edu /icip/itm/Lessonpl/calc/portos/perfect.html (371 words)

	QBasic - number series, bases & manipulation (Site not responding. Last check: )
		The series of numbers that he calculated was 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
		This is the series of numbers which are equal to the sum of all of its positive divisors, excluding itself.
		All the variations for that number are 5678, 5687, 5768, 5786, 5867, 5885, 6578, 6587, 6758, 6785, 6857, 6875, 7568, 7586, 7658, 7685, 7867, 7876, 8567, 8576, 8657, 8675, 8756 and 8765.
	www.brisray.com /qbasic/qnumber.htm (2974 words)

	An extension of the results of Servais and Cramer on odd perfect and odd multiply perfect numbers American Mathematical ... (Site not responding. Last check: )
		An odd perfect number is defined to be an odd integer that is equal to the sum of its proper divisors.
		An odd multiply perfect (or multiperfect) number is an odd number the sum of whose proper divisors is equal to an integral multiple (>=2) of the number.
		Specifically, he proved that if n is the number of distinct prime divisors of an odd perfect number, then its least prime divisor does not exceed n.
	www.findarticles.com /p/articles/mi_qa3742/is_200301/ai_n9227471 (466 words)

	Numerology in Religion
		The odd number "3" which is that of the Trinity is a very strong number and certainly Jewish numerology recognizes it as such.
		The odd number "7" represents the perfect number of the Hebrew Scriptures and means "fullness." The Church has seven sacraments connoting that its mystical way imparts the fullness of Divine Grace.
		The number "9" is a "super-strong" number as it is "3 x 3." The number of fish caught by the Apostles after the Resurrection of Christ is noted in John's Gospel as "153." To add these three numbers is to come up with the number "9."
	www.unicorne.org /orthodoxy/jan2003/funerals.htm (293 words)

	Perfect-Key
		numbered column from the left where ‘n’ equals the base number of the previous column.
		numbered column from the left where ‘n’ equals the exponent of previous column.
		numbered column from the left, apply 4(n+1)-1 where ‘n’ equals the number of the previous column.
	www.borderschess.org /perfect.htm (655 words)

	[No title] (Site not responding. Last check: )
		But it's beyond current mathematics to prove that there are no more; in particular, it would include solving the ancient problem of whether odd perfect numbers exist (since twice an odd perfect would be one of these "triperfect" numbers).
		In particular, showing that there are no more triperfect numbers than the six currently known would also prove that there are no odd perfect numbers, since twice an odd perfect number is triperfect.
		It is the case that some of those investigating multiperfect numbers feel quite confident that all of index 3, 4, 5 and 6 have been found, but there are no proofs to back these assertions.
	www.math.niu.edu /~rusin/known-math/99/multperf (298 words)

	F_rcase.html
		The most recent record for the number of perfect numbers that has been found is 34.
		But a number q with a single proper divisor d must be a prime so d must equal 1.
		B.C. The characterization of even perfect numbers is thanks to Euler sometime in the eighteenth century.
	math.ucsd.edu /~jwavrik/proj107b/F_rcase3.html (489 words)

	Number Patterns, Curves & Topology
		Binary numbers use the same rules as decimal numbers, that is, the value of any digit (bit) depends on its position in the whole number.
		Use the arrows or the slider bar to explore the relationship between decimal and binary numbers from 0 to 255.
		The same author's Mystic Rose features a variable number of points evenly spaced around a circle in which every point is joined to every other point.
	britton.disted.camosun.bc.ca /jbfunpatt.htm (7294 words)

	Numbers
		Abundant numbers are numbers whose factors add up to more than the number.
		Complex numbers are in the form a+bi, where i is the square root of negative one.
		There are very few perfect numbers, this is because all perfect numbers are the sum of all the numbers from one to a mersenne prime.
	campus.fortunecity.com /newton/970/numbers.html (859 words)

	Number Session 1: Solutions
		All even numbers are multiples of 2 and appear in the row and column for 2 in the multiplication table.
		According to the table, every number is a multiple of 3, which may seem surprising; what it means, though, is that any units digit can be the result when we multiply by 3.
		Problem H7 The perfect squares are the numbers on the main diagonal of the multiplication table (0 • 0, 1 • 1, 2 • 2, etc.).
	www.learner.org /channel/courses/learningmath/number/session1/solutions_homework.html (661 words)

	Occultists Worship Numbers
		Further, the number 6 + 21 = 27, another number of power, because it is formed by the multiplication of 3x9.
		When eleven is multiplied by the perfect number 3, the number 33 is produced, a number of tremendous occult importance.
		Numbers 11 and 22 are sacred primary occult numbers.
	www.cuttingedge.org /pages/seminar2/NUMBERS.htm (1216 words)

	Perfect number - Wikipedia, the free encyclopedia
		Curtiss (1922) uses a greedy algorithm for Egyptian fractions to prove that a perfect number N must have a number of divisors at least proportional to lnlnN.
		A much stronger singly-logarithmic bound would follow from the nonexistence of odd perfect numbers and the known form of even perfect numbers.
		McDaniel, The non-existence of odd perfect numbers of a certain form, Archiv der Mathematik (Basel), vol.
	en.wikipedia.org /wiki/Perfect_number (1372 words)

	Science: Multiperfect numbers proliferate in Colorado - 01 May 1993 - New Scientist (Site not responding. Last check: )
		A number is said to be perfect if the sum of its proper divisors equals the number itself.
		For example, 6 is a perfect number because its proper divisors are 1, 2 and 3.
		Although perfect numbers and their multiply-perfect cousins serve no practical purpose, they have amused mathematicians and philosophers from Euler to Descartes.
	www.newscientist.com /article/mg13818713.200-science-multiperfect-numbers-proliferate-in-colorado-.html (277 words)

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