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Topic: Pentagonal number theorem

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	pentagonal number (Site not responding. Last check: 2007-10-09)
		A pentagonal number is a figurate number that represents a pentagon.
		Pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.
		The nth pentagonal number is one third of the 3n-1th triangular number.
	www.abacci.com /wikipedia/topic.aspx?cur_title=pentagonal_number (215 words)

	NationMaster - Encyclopedia: Pentagonal number
		Pentagonal numbers should not be confused with centered pentagonal numbers.
		Therefore the number of dots in the parallelogram is n(n+1), and the number of dots in the triangle is
		Likewise in the case of square numbers, one can visualize that pentagonal numbers are constituted with triangular numbers: in particular, this sketch presents a pentagonal number of rank 5 as the sum of a triangular number of rank 5 and two triangular numbers of rank 4.
	www.nationmaster.com /encyclopedia/Pentagonal-number (357 words)

	Pentagonal number - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-09)
		A pentagonal number is a figurate number that represents a pentagon.
		Pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.
		The nth pentagonal number is one third of the 3n-1th triangular number.
	en.wikipedia.org /wiki/Pentagonal_number (183 words)

	NationMaster - Encyclopedia: Pentagonal number theorem
		In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
		60 is the smallest number divisible by the numbers 1 to 6.
		70 is a Pell number and a Harshad number.
	www.nationmaster.com /encyclopedia/Pentagonal-number-theorem (1468 words)

	PlanetMath: polygonal number
		Polygonal numbers were studied somewhat by the ancients, as far back as the Pythagoreans, but nowadays their interest is mostly historical, in connection with this famous result:
		In other words, any nonnegative integer is a sum of three triangular numbers, four squares, five pentagonal numbers, and so on.
		This is version 2 of polygonal number, born on 2003-09-02, modified 2003-09-03.
	planetmath.org /encyclopedia/PentagonalNumber.html (216 words)

	Pentagonal number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		A pentagonal number is a (Click link for more info and facts about figurate number) figurate number that represents a (The United States military establishment) pentagon.
		Pentagonal numbers are important to (Swiss mathematician (1707-1783)) Euler's theory of partitions, as expressed in his (Click link for more info and facts about pentagonal number theorem) pentagonal number theorem.
		Pentagonal numbers should not be confused with (Click link for more info and facts about centered pentagonal number) centered pentagonal numbers.
	www.absoluteastronomy.com /encyclopedia/p/pe/pentagonal_number.htm (206 words)

	Pentagonal number theorem - Wikipedia, the free encyclopedia
		The theorem can be given a combinatorial interpretation in terms of partitions.
		In particular, the left hand side is a generating function for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts.
		In summary, we have shown that partitions into an even number of distinct parts and an odd number of distinct parts exactly cancel each other out, except for pentagonal numbers, where there is exactly one case left over (which contributes a factor of (-1)
	en.wikipedia.org /wiki/Pentagonal_number_theorem (803 words)

	Permutations by Inversion
		The unimodal behavior of the inversion numbers suggests that the number of inversions in a random permutation may be asymptotically normal.
		Theorem 3 [Knuth, Netto][8],[11], The inversion numbers I
		are the pentagonal numbers, sequence A000326 in [13],
	academic.csuohio.edu /bmargolius/homepage/inversions/invers.htm (1404 words)

	Figurate number Summary
		These numbers were studied, as were many kinds of numbers, for the sake of their supposed mystical properties rather than for their practical value.
		Figurate numbers were a concern of Pythagorean geometry, since Pythagoras is credited with initiating them, and the notion that these numbers are generated from a gnomon or basic unit.
		The tedium of increasing number of subtractions as the number grows is bypassed by a method similar to the standard way of square-rooting taught in school.
	www.bookrags.com /Figurate_number (2678 words)

	Pentagonal number theorem - Encyclopedia Glossary Meaning Explanation Pentagonal number theorem (Site not responding. Last check: 2007-10-09)
		Pentagonal number theorem - Encyclopedia Glossary Meaning Explanation Pentagonal number theorem.
		In particular, the left hand side is a generating function (for similar reasons as the generating function for the more generalized unrestricted partition function) for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts.
		The modulus of the Euler function (see Q-series for picture) shows the fractal modular group symmetry and occurs in the study of the interior of the Mandelbrot set.
	www.encyclopedia-glossary.com /en/Pentagonal-number-theorem.html (686 words)

	PlanetMath: pentagonal number theorem
		The theorem was discovered and proved by Euler around 1750.
		Cross-references: generating functions, functions, Euler, pentagonal numbers, exponents, bijection, integer, right, disjoint, union, cardinal, mapping, natural number, odd, even, sum, partitions, product, formal power series, sides
		This is version 4 of pentagonal number theorem, born on 2003-09-15, modified 2003-10-19.
	planetmath.org /encyclopedia/PentagonalNumbersTheorem.html (186 words)

	[No title]
		Consequence: The number of partitions of n into parts of size at most A is equal to the coefficient of x^n in the formal power series 1/(1-q)(1-q^2)...(1-q^A).
		The coefficient of x^n in (1+x^2)/(1—x^3) is the number of partitions of n into parts of size 2 and 3, where the part of size 2 cannot be repeated.
		Franklin’s bijective proof: Assign a partition with distinct part weight (-1)^k, where k is the number of parts, so that the coefficient of x^n in the LHS is the sum of the weights of the partitions of n into distinct parts.
	www.math.wisc.edu /~propp/192/11-06.doc (1015 words)

	Shop Fresh : Article 'Universal coefficient theorem' (Site not responding. Last check: 2007-10-09)
		In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense determine the groups Hi(X,A).
		The universal coefficient theorem explains that homology with integer coefficients determines all other homology theories, by use of the tensor product; it is not anodyne, in that (as we would now put it) the tensor product has derived functors that enter into a general formulation.
		This is an unsolved problem, as of 2004; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.
	www.shop-fresh.net /DisplayArticle428121.html (745 words)

	The On-Line Encyclopedia of Integer Sequences
		Number of levels in the partitions of n+1 with parts in {1,2}.
		Sequence consists of the pentagonal numbers (A000326), followed by A000326(n)+n, and then the next pentagonal numbers.
		A000326 (pentagonal numbers), A000217 (triangular numbers), A010815, A034828, A000326, A005449.
	www.research.att.com /cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001318 (483 words)

	Pentagonal number theorem
		In mathematics, the pentagonal number theorem, originally due to Euler, states that
		In particular, the left hand side is a generating function (for similar reasons as the generating function for the more generalized unrestricted partition function) for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts.
		This gives a beautiful recurrence for calculating, the number of partitions of n Partition function (number theory).
	www.xasa.com /wiki/en/wikipedia/p/pe/pentagonal_number_theorem.html (646 words)

	ipedia.com: Partition function (number theory) Article (Site not responding. Last check: 2007-10-09)
		The partition function p is a non-multiplicative function and represents the number of possible partitions of a natural number n, which is to say the number of distinct ways of representing n as a sum...
		The partition function p(n) is a non-multiplicative function and represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers.
		The number of partitions meeting the second condition is p(k+1,n) since a partition into parts of at least k which contains no parts of at exactly k must have all parts at least k+1.
	www.ipedia.com /partition_function__number_theory_.html (487 words)

	Read about Category:Number theory at WorldVillage Encyclopedia. Research Category:Number theory and learn about ... (Site not responding. Last check: 2007-10-09)
		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)

	Combinatorics/Partitions Seminar
		In the process, we also prove a number of other congruences modulo 2 (and, for free, we provide another proof of the modulo 3 congruence that they highlight in their paper).
		This linear combination counts the number of partitions of the set of m x n word representations that are inequivalent under D_2.
		Integer analogs of lecture hall theorems and the combinatorics of l-sequences.
	www.math.psu.edu /keith/Combinatorics-PartitionsFall2006.html (779 words)

	Gresham College \| Search Lectures and Events
		This number occurs throughout statistics for example, if everyone in this room were to write down ten pairs of numbers, then the proportion of all those pairs that have no common factor would be very close to its reciprocal, 6/p2.
		Euler proved, using his generating functions, that for any number, the number of odd partitions is always equal to the number of distinct partitions an intriguing and unexpected result.
		Fermats little theorem states that, if p is a prime number, and a is any number that is not divisible by p, then ap1 1 must be divisible by p.
	www.gresham.ac.uk /event.asp?PageId=4&EventId=67 (3622 words)

	Interactive Mathematics Miscellany and Puzzles
		It's a pentagonal knot that used to tile the background of this and subsequent pages.
		But a Lemma and a Theorem and a very real proof that exploits well known properties of parallel lines are still available.
		One page currently presents 72 different proofs of the Pythagorean Theorem which was a great fun putting together.
	www.cut-the-knot.org /front.shtml (1132 words)

	Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics) - Books
		This introductory textbook is designed to teach undergraduates the basic ideas and techniques of number theory, with special consideration to the principles of analytic number theory.
		As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours.
		The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n).
	www.wenshop.com /detail/0387901639.html (1192 words)

	Movies.com: Marketplace
		He gives a reference to An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright for the detailed proof, but the reader may be required to do unnecessary guessing (which is believed to be good in learning math, but seems to be nothing but trouble for me) to go through it.
		As a second text on number theory, and an introduction to the aspects of number theory related to function theory and analysis, I believe that Apostol's book is the best that anyone could possibly hope for.
		As mentioned somewhere in the book, one of the aims of the author is to arouse reader's interest in number theory..which this book will certainly do..especially since its main emphasis is on prime numbers.
	movies.go.com /marketplace/details?asin=0387901639&allreviews=true (1361 words)

	Matches for:
		The subject of $q$-series can be said to begin with Euler and his pentagonal number theorem.
		In 1940, G. Hardy described what we now call Ramanujan's famous $_1\psi_1$ summation theorem as "a remarkable formula with many parameters." This is now one of the fundamental theorems of the subject.
		The excellent lectures that are included chart pathways into the future and survey the numerous applications of $q$-series to combinatorics, number theory, and physics.
	www.mathaware.org /bookstore?fn=20&arg1=conmseries&item=CONM-291 (335 words)

	Amazon.com: Number Theory in the Spirit of Ramanujan: Books: Bruce C. Berndt (Site not responding. Last check: 2007-10-09)
		Ramanujan is recognized as one of the great number theorists of the twentieth century.
		This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function.
		Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan.
	www.amazon.com /Number-Theory-Spirit-Ramanujan-Berndt/dp/0821841785 (1020 words)

	Contribution (Site not responding. Last check: 2007-10-09)
		Pentagonal numbers are numbers of the form n(3n-1)/2, where n is a natural number.
		The generalized pentagonal numbers are of the same form, but with n as integers.
		It is easy to show that the generalized pentagonal number is of form n(3n-1)/2 or n(3n+1)/2.
	euler-life.org /cont.html (56 words)

	A Pentagonal Number Sieve - Corteel, Savage, Wilf, Zeilberger (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		A Pentagonal Number Sieve - Corteel, Savage, Wilf, Zeilberger (ResearchIndex)
		Abstract: We prove a general "pentagonal sieve" theorem that has corollaries such as the following.
		Theorem 9 In a prefab P, let f m (n) be the number of m tuples of objects of order n in P such that no prime object is a...
	citeseer.ist.psu.edu /corteel98pentagonal.html (386 words)

	Students - Research - MS 497 Proposals - 2001 - Mathematics and Computer Science, Stetson University (Site not responding. Last check: 2007-10-09)
		ABSTRACT: An interpretation of Euler's Pentagonal Number Theorem states that the number of partitions of a positive integer n into an even number of distinct parts is equinumerous with the number of partitions of n into an odd number of distinct parts, except when n is a pentagonal number.
		In this research, we seek a combinatorial proof of an analogue of Euler's Pentagonal Number Theorem.
		ABSTRACT: Just as a regular graph is one in which each vertex is distance 1 away from exactly the same number of vertices, a semiregular graph is a graph in which each vertex is distance 2 away from exactly the same number of vertices.
	www.stetson.edu /departments/mathcs/students/research/math/ms498/2001/index.shtml (430 words)

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