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Topic: Bernoulli numbers

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	PlanetMath: Bernoulli number
		and, in fact, the Bernoulli numbers are usually defined as the coefficients that appear in such expansion.
		These combinatorial numbers occur in a number of contexts; the most elementary is perhaps that they occur in the formulas for the sum of the
		This is version 8 of Bernoulli number, born on 2001-10-15, modified 2006-11-29.
	planetmath.org /encyclopedia/BernoulliNumber.html (164 words)

	Bernoulli number - Wikipedia, the free encyclopedia
		In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory.
		The Bernoulli numbers may also be defined using the technique of generating functions.
		Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny-Artin-Chowla.
	en.wikipedia.org /wiki/Bernoulli_number (1035 words)

	Bernoulli_Jacob biography
		Jacob Bernoulli was the brother of Johann Bernoulli and the uncle of Daniel Bernoulli.
		Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus...
		Johann Bernoulli's boasts were the first cause of Jacob's attacks on him and Jacob wrote that Johann was his pupil whose only achievements were to repeat what his teacher had taught him.
	www-history.mcs.st-andrews.ac.uk /Biographies/Bernoulli_Jacob.html (1894 words)

	Jakob Bernoulli Summary
		Bernoulli's pamphlet on parallels of logic and algebra appeared in 1685, again furthering his reputation, and in 1687 he was offered a post at the University of Basel as a professor of mathematics.
		Bernoulli was fascinated by the mathematical properties of curves, especially the logarithmic spiral, a figure similar to the chambered nautilus mollusk shell in nature with its perfectly symmetrical spirals.
		Bernoulli formulated an equation to refute the traditional hypothesis among mathematicians that the catenary is a parabola.
	www.bookrags.com /Jakob_Bernoulli (3052 words)

	Introduction on Bernoulli's numbers
		Bernoulli's numbers play an important and quite mysterious role in mathematics and in various places like analysis, number theory and differential topology.
		Perhaps one of the most important result is Euler-Maclaurin summation formula, where Bernoulli's numbers are contained and which allows to accelerate the computation of slow converging series (see the essay on Euler's constant at [9]).
		According to Louis Saalschültz [17], the term Bernoulli's numbers was used for the first time by Abraham De Moivre (1667-1754) and also by Leonhard Euler (1707-1783) in 1755.
	numbers.computation.free.fr /Constants/Miscellaneous/bernoulli.html (1028 words)

	Biographies
		Bernoulli's parents became promiment members of Basel society and compelled their eldest son Jakob to study theology and philosophy.
		Bernoulli's younger brother Johann was initially forced to study medicine but followed in Jakob's footsteps and studied mathematics and physics instead.
		Jakob Bernoulli held the chair in mathematics at Basel University until his death in 1705 when he was succeeded by his brother Johann who had long coveted the position.
	tulsagrad.ou.edu /statistics/biographies/Bernoulli.htm (417 words)

	Bernoulli
		Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687.
		Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning.
		Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability.
	www.thocp.net /biographies/bernoulli.html (1926 words)

	Bernoulli numbers and the Pascal triangle
		The Bernoulli numbers plays an important role in mathematics.They first appeard in Ars Conjectandi, a famous (and posthumous) published treatise in 1713, by Jakob Bernoulli.
		Bernoulli`s numbers appears in analysis, number theory and differential topology.
		Thus, the Bernoulli numbers may be determined step by step from the last symbolic formula; note that after the binomial expansion the powers of the B numbers must be replaced by Bernoulli numbers with the appropriate indices.
	milan.milanovic.org /math/english/bernoulli/bernoulli.html (235 words)

	Contents
		The triangular number is a polygonal number: a number that can be represented by a regular geometric arrangement of equally spaced points.
		Tetrahedral numbers are pyramidal numbers and are the sum of consectuive triangular numbers.
		The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.Fibonacci slides.
	milan.milanovic.org /math/english/contents.html (1810 words)

	Bernoulli Number
		These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.
		(Sloane's A068399), while the numbers of digits in the corresponding denominators are 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 4, 1, 3, 5, 3,...
		Bernoulli numbers appear in expressions of the form
	www.m-brella.be /math/topics/BernoulliNumber.html (884 words)

	Bernoulli Bibliography
		Bernoulli numbers are particularly important in number theory, especially in connection with Fermat's last theorem (see, e.g., Ribenboim (1979)).
		The number sequences of Euler, Genocchi, Stirling and others, as well as the tangent numbers, secant numbers, etc., are closely related to the Bernoulli numbers.
		The same is true for the numerous generalizations and extensions of the Bernoulli and allied numbers and of the corresponding polynomials.
	www.mscs.dal.ca /~dilcher/bernoulli.html (997 words)

	Numeracy + Computer Literacy (1 of 4)
		If we have N spheres, and N is a triangular number, then these spheres may be shaped into a triangle with the same number of spheres to a side, as shown at right.
		The Bernoulli numbers crop up all over in mathematics, including in the coefficients of polynomials of degree n+1 giving the sum of consecutive integers to the nth power.
		Her paper on how the Babbage difference engine might be used to compute Bernoulli numbers earned her the title of "first computer programmer" and the USA Defense Department named the Ada computer language in her honor in 1980.
	www.4dsolutions.net /ocn/numeracy0.html (3159 words)

	Bernoulli Numbers, B(n) -- from Harry J. Smith
		Bernoulli Numbers, B(n), are rational numbers and are defined for all n >= 0.
		The exact numerator and denominator of all even indexed Bernoulli numbers up to B(x) are generated and saved in computer storage.
		This function returns the exact numerator of the Bernoulli number B(x), a rational number reduced to its lowest terms.
	www.geocities.com /hjsmithh/Numbers/Bernoulli.html (311 words)

	Jacob Bernoulli
		During the time that Jacob Bernoulli was taking his university degree in theology he was studying mathematics and astronomy against the wishes of his parents.
		Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus.
		In 1690 in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation.
	www.stetson.edu /~efriedma/periodictable/html/Br.html (785 words)

	Sums of Consecutive Powers
		In general, we may suspect that the sum of the first natural numbers raised to the pth power is a polynomial in n of degree p + 1.
		Bernoulli noted that since the Bernoulli polynomials are defined by (3) up to a constant and
		Bernoulli numbers in sums of powers, power series, and the Riemann zeta function
	www.math.rutgers.edu /~erowland/sumsofpowers.html (670 words)

	Open Questions: The Riemann Hypothesis
		Now, whenever a number theorist deals with functions defined on (positive) integers, one of the first impulses is to think about the Dirichlet series which is the "generating function" for the given sequence of numbers.
		This class number is number of elements of the "ideal class group" of the field.
		The prime number theorem, the Riemann hypothesis, the functional equation of the zeta function, and the abc conjecture are the topics of interest with respect to number theory.
	www.openquestions.com /oq-ma014.htm (14197 words)

	Eulerian Numbers
		Thus the numbers of permutations having exactly 0, 1, and 2 rises are 1, 4, and 1 respectively, and these numbers comprise the 3rd row of Eulerian numbers.
		There's also a nice relation between Eulerian numbers and the generalized Bernoulli numbers. Let W(m,n) be a weighted average of the nth powers of the first m natural numbers, using the Eulerians as the weights. For example
		Interestingly, this formula also generates the Eulerian numbers, so both sets of numbers can be regarded as simply different regions of a single array. Here's a brief table of the combined binomial and Eulerian numbers.
	www.mathpages.com /home/kmath012/kmath012.htm (840 words)

	[No title]
		Nxx denotes the absolute value of the numerator of the Bernoulii number B_xx.
		Pxx means a prime number with xx decimal digits.
		A "$" at the end of a line means the number continues on the next line.
	www.cerias.purdue.edu /homes/ssw/bernoulli/bnum (804 words)

	Bernoulli Numbers and Harmonic Series (Site not responding. Last check: 2007-11-03)
		Interestingly, it appears that the discrepancy between H(n) and log(n)+gamma can be computed very directly from the Bernoulli numbers.
		When I said the summation _appears_ to converge very rapidly, I was referring to the fact that the terms initially become extremely small as k increases.
		But in the long run the values of the Bernoulli numbers increase at a super-exponential rate, as can be seen from the inequality 2(2k)!
	www.mathpages.com /home/kmath284.htm (577 words)

	Biography of Johann Bernoulli
		The Bernoulli family was probably the most notable mathematical family in world history.
		The modern Bernoulli numbers are a superset of the old (archaic) version.
		Bernoulli also took an aspiring young, well known, intellectual by the name of Leonhard Euler and tutored him for quite a while.
	www.andrews.edu /~calkins/math/biograph/bioberno.htm (629 words)

	Jakob Bernoulli (Site not responding. Last check: 2007-11-03)
		Although Jacob and Johann Bernoulli both worked on similar problems their relationship was soon to change from one of collaborators to one of rivals.
		Johann Bernoulli would have liked the chair of mathematics at Basel which Jacob held and he certainly resented having to move to Holland in 1695.
		The first two of these contained many results, such as fundamental result that S(1/n) diverges, which Bernoulli believed were new but they had actually been proved by Mengoli 40 years earlier.
	www.mathematik.ch /mathematiker/jakob_bernoulli.php (1815 words)

	Topic of Bernoulli's Number (Site not responding. Last check: 2007-11-03)
		Bernoulli's Number one is equal to negative one-half.
		The most relevant use of Bernoulli Numbers are found in Fermat's Last Theorem as expressed in the von-Staudt-Clausen Theorem.
		All odd Bernoulli Numbers (except one) are equal to zero.
	www.andrews.edu /~calkins/math/biograph/199899/topbern.htm (197 words)

	[No title]
		Mapping Class Groups, Characteristic Classes and Bernoulli Numbers GUIDO MISLIN ETH Z"urich, Switzerland; and Ohio-State University, Columbus, Ohio Introduction The mapping class group g of a closed, connected and oriented surface Sg of genus g is defined as the group of connected components of the group of orienta* *tion preserving diffeomorphisms of Sg.
		The characteristic classes are related to the denominator* s of Bernoulli* numbers, the Euler characteristic involves the whole Bernoulli number* s, and our theorems concerning the Yagita invariant have to do with the notion of regular primes, which is expressible in terms of numerators of Bernoulli* number* *s.
		The Bernoulli numbers Bn are rational numbers defined recursively by the formula (B + 1)#n - Bn = 0 ; n 2 ; where the exponent "# n" means that after evaluating the n'th power of the mono- mial, one replaces the power Bk by Bk.
	hopf.math.purdue.edu /Mislin/bernoulli.txt (11294 words)

	Combinatorial Functions
		Harmonic numbers appear in many combinatorial estimation problems, often playing the role of discrete analogs of logarithms.
		Numerical values for Bernoulli numbers are needed in many numerical algorithms.
		This gives the number of partitions of 100, with and without the constraint that the terms should be distinct.
	documents.wolfram.com /v4/MainBook/3.2.5.html (410 words)

	Math 304: Bernoulli's formula (Site not responding. Last check: 2007-11-03)
		In fact, computing powers directly is more efficient (fewer operations) when N is small compared to n, and Bernoulli's formula is more efficient when n is small compared to N.
		You need to compute two sets of numbers recursively, the binomial coefficients and the Bernoulli numbers.
		in the recursion formula for Bernoulli numbers means to insert terms using the same pattern.
	www.case.edu /artsci/math/alexander/math304/bernoulli.html (280 words)

	James Bernoulli - Wikipedia, the free encyclopedia
		James Bernoulli (also known as Jacob I) was born in Basel, Switzerland on December 27, 1654 and lived until August 16, 1705.
		He is one of the eight prominent mathematicians in the Bernoulli family.
		This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers.
	en.wikipedia.org /wiki/Jakob_Bernoulli (310 words)

	Sums of Consecutive Powers Project
		There is a story in the mathematical folklore about the great mathematician Gauss (1777–1855), who, while in elementary school, reportedly found the sum of the first hundred natural numbers in a matter of minutes.
		After all, we have only changed the problem of computing sums of like powers into one of finding the coefficients of Bernoulli polynomials, and we don't have a closed-form expression for these coefficients.
		There is a hierarchy of mathematical objects that appear in nature, and the Bernoulli polynomials (and especially the Bernoulli numbers) happen to lie outside of any "direct" construction from the integers.
	www.math.rutgers.edu /~erowland/sumsofpowers-project.html (1333 words)

	The Prime Glossary: Bernoulli number (Site not responding. Last check: 2007-11-03)
		The Bernoulli numbers come from the coefficients in the Taylor expansion of x/(e
		These numbers can also be defined using the Riemann zeta function as follows
		The Bernoulli numbers first appeared in the posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli.
	primes.utm.edu /glossary/page.php?sort=BernoulliNumber (108 words)

	Bernoulli and Euler numbers (Site not responding. Last check: 2007-11-03)
		This site contains the full version of a paper, "Prime divisors of the Bernoulli and Euler numbers," whose abbreviated version was published in the Proceedings of the Millennial Conference on Number Theory, held at the University of Illinois, Urbana, Illinois, May 21--26, 2000.
		The paper appears on pages 357--374 of volume III of Number Theory for the Millennium, A K Peters, 2002.
		They give the known prime factors of the Bernoulli numerators with subscript up to 300, and those of the Euler numbers with subscript up to 200.
	homes.cerias.purdue.edu /~ssw/bernoulli/index.html (288 words)

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