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Topic: Euler numbers

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	Euler (Site not responding. Last check: 2007-09-07)
		Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair.
		Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772.
		Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Euler.html (4046 words)

	Euler Number
		The Euler Number of a surface is an integer with this properties:
		Therefore, in order to prove that the results of computing the Euler Number for different divisions yield equal results, it is suffice to prove this for the case in which one division is a refinement of the other.
		The fact that the Euler Numbers of the Torus and the Sphere are not equal shows that these objects are not topologically equivalent.
	www.math.toronto.edu /~drorbn/People/Eldar/thesis/euler_char.htm (1032 words)

	Euler pseudoprime - Wikipedia, the free encyclopedia
		It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.
		The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 1729 = 7·13·19.
		A discussion of numbers of this form can be found at Euler-Jacobi pseudoprime.
	en.wikipedia.org /wiki/Euler_pseudoprime (282 words)

	10.9. Euler, Leonard (1707-1783)
		Leonhard Euler was one of top mathematicians of the eighteenth century and the greatest mathematician to come out of Switzerland.
		Euler took up a position at the Academy of Sciences in St. Petersburg, Russia, in 1727 and became the professor of mathematics six years later.
		Unfortunately, Euler's philosophical ability was limited and he often blundered to the amusement of all involved.
	web01.shu.edu /projects/reals/history/euler.html (906 words)

	LEONHARD EULER (Site not responding. Last check: 2007-09-07)
		Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century (His closest competitor for that title is Lagrange) and one of the most prolific of all time; his publication list of 886 papers and books may be exceeded only by Paul Erdös.
		But there is another "Euler's formula" that (to use the modern terminology adopted long after Euler's death) gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers.
		Euler's study of the bridges of Königsberg can be seen as the beginning of combinatorial topology (which is why the Euler characteristic bears his name).
	www.usna.edu /Users/math/meh/euler.html (402 words)

	Mathematics Magazine: Higher trigonometry, hyperreal numbers, and Euler's analysis of infinities (Site not responding. Last check: 2007-09-07)
		Because of this, Euler is often portrayed in popular accounts and classroom lectures as a reckless symbol-manipulator, who worked in a number system fraught with nonsense and contradiction, but who through sheer intuitive brilliance somehow came to correct conclusions.
		The first requirement of our rehabilitation of Euler's arguments is that his methods be formulated within a mathematical system in which the properties of infinite and infinitesimal numbers are explained at least as clearly as the properties of the real numbers.
		The real numbers are described informally in the main body of the textbook, but presented more precisely in an appendix by citing the field axioms, the order axioms, the definition of the natural numbers, the root axiom (that principal nth roots exist for positive numbers), and the completeness axiom.
	www.findarticles.com /p/articles/mi_qa3789/is_200112/ai_n9012342 (1437 words)

	Biography of Leonhard Euler
		Leonhard Euler, from Basel, Switzerland, was one of the most prolific mathematicians of all time.
		It was once said that "Euler calculated without apparent effort, as men breathe..." Although he was set back by partial blindness in his 20's, and then almost complete blindness in his later life, he was still able to discover some of the most important mathematical works of his time.
		Euler claimed that while holding a baby in his arms, and children playing around his feet, he made some of his best mathematical discoveries.
	www.andrews.edu /~calkins/math/biograph/199899/bioeuler.htm (1119 words)

	Index to On-Line Encyclopedia of Integer Sequences
		Euler transforms: (1) A000070 A000097 A000098 A000237 A000335 A000391 A000417 A000428 A000608 A000710 A000711 A000712
		Euler transforms: (2) A000713 A000714 A000715 A000716 A001372 A001373 A001384 A001385 A001970 A003080 A003094 A004101
		Euler transforms: (3) A004113 A005470 A005750 A007003 A007441 A007562 A007563 A007713 A007714 A007864 A018243 A023871
	www.research.att.com /~njas/sequences/Sindx_Eu.html (367 words)

	[No title]
		Euler's Constant was first introduced by Leonhard Euler (1707-1783) in 1734 as
		Direct use of formula (1) to compute Euler constant is of poor interest since the convergence is very slow.
		Integral and series formulae for the Euler constant can be found in Collection of formulae for the Euler constant.
	numbers.computation.free.fr /Constants/Gamma/gamma.html (1154 words)

	Perfect numbers (Site not responding. Last check: 2007-09-07)
		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.
		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.
	www-history.mcs.st-and.ac.uk /history/HistTopics/Perfect_numbers.html (4298 words)

	Floating-Point Numbers
		For a negative number, we may set the sign bit of the floating-point word and negate the number to be encoded, leaving only nonnegative numbers to be considered.
		The basic idea of floating point encoding of a binary number is to normalize the number by shifting the bits either left or right until the shifted result lies between 1/2 and 1.
		The normalized number between 1/2 and 1 is called the significand, so called because it holds all the ``significant bits'' of the number.
	ccrma.stanford.edu /~jos/euler/Floating_Point_Numbers.html (503 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-09-07)
		"Whoa," we said, "that's no number I've ever seen before." Of course, we really had seen it before, in folk tales and legends and when our big sisters brought home their calculus homework, but this was the first time we'd really seen it in a math class.
		He coined the word Logarithm, which means "number of the ratio", as in the common ratio of a geometric sequence.
		Anyway, e and Pi are both numbers that will pop out of your problems when you least expect it, and I'd say that they do it with about the same frequency.
	forum.swarthmore.edu /dr.math/problems/e_and_pi_2.html (971 words)

	Prime numbers
		One of the major contribution of Euler was introduction of Gamma function.
		The non-zero Euler number are odd integers which alternate in sign.
		These Euler numbers, along with the Bernoulli numbers are helpful when attempting to approximate the Taylor series for complex functions involving the secant, tangent, and others that do not have simple formulas.
	www.math.uri.edu /~kulenm/diffeqaturi/m381f00fp/barbara/barbaramp.html (712 words)

	Euler Numbers
		EN is a library function for computing Euler numbers and polynomials.
		The EULER NUMBERS command can be used to generate a sequence of Euler numbers.
		Typically and are 1 and is set to the number of Euler numbers to generate.
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/eulenumb.htm (222 words)

	Noticing Numbers
		Examination of these sequences in relation to shells of power values (also called nexus numbers or difference series) and in conjunction with sequences of power values and sequences of summations of powers, might nicely inform much more sophisticated mathematics than mentioned on these pages.
		One equation is an extension of Euler's Triangle, the other is an extension of Worpitzky's Identity.
		She also thinks harmony in number is just as interesting as dissonance which (is emphasized in prime-number studies).
	noticingnumbers.net (509 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-09-07)
		In this paper we derive a formula for the Euler numbers of four-dimensional rotating fl holes by the integral of the Euler density on the spacetime manifolds of fl holes.
		Using this formula, we obtain that the Euler numbers of Kerr and Kerr-Newman fl holes are 2.
		We also obtain that the Euler number of the Kerr-Sen metric in the heterotic string theory with one boost angle nonzero is 2, which is in accordance with its topology.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20505923 (253 words)

	April 15: Mathematician Leonhard Euler born
		Euler showed early promise in math, a promise that was amply fulfilled.
		The Euler constant, Euler numbers, Eulerian integrals and other mathematical forms are named for him.
		But she died before the job was given to him and Euler was almost forced to accept a position in the Russian navy to support himself.
	chi.gospelcom.net /DAILYF/2002/04/daily-04-15-2002.shtml (613 words)

	Introduction on Bernoulli's numbers
		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]).
		They also appear in numbers theory (Fermat's theorem) and in many other domains and have caused the creation of a huge literature (see the 2700 and more entries enumerated in [6]).
		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)

	Imaginary Numbers
		When negative numbers are multiplied, we are given certain rules for the operation: (1) When two positive numbers are multiplied, the result is positive; (2) when two negative numbers are multiplied, the result is also positive; and (3) when a negative and a positive number are multiplied, the result is negative.
		The only number that is neither positive nor negative is zero, and zero multiplied by anything (positive, negative, or zero) is zero [note].
		That there actually are no imaginary numbers now does not mean that they are nothing: They exist conceptually, or intentionally, in representation, which is as much reality as they need before some operation produces real numbers and real results.
	www.friesian.com /imagine.htm (3689 words)

	euler and complex numbers (euler and complex numbrs) information.
		A complex number is a number with a real and an imaginary part,..
		A very significant result in the area of complex numbers is Euler's theorem.
		Complex Numbers are numbers of the form a + bi, where a and b are Real Numbers and..
	www.wide-find.com /e/euler_and_complex_numbers.html (196 words)

	Euler-Pascal/Cube
		The algorithm that generates formulas for power values (that is Worpitzky's Identity of 1883) logically also generates formulas for values that are summations of powers (where Bernoulli numbers come into play) as well as shells or nexus number series and more.
		In order to generate sequences that are Euler Triangle rows (see A008292) or accumulations from Euler Triangle rows, values of Euler/Pascal cube series are defined with the variables of nth power level, kth order of occurrence and zth accumulation level.
		The number of summations from an initial Euler Triangle row is “enumerated” by the value that is z.
	noticingnumbers.net /300SeriesCube.htm (398 words)

	The 15 Most Famous Transcendental Numbers - Cliff Pickover
		Even so, only a few classes of transcendental numbers are known to humans, and it's very difficult to prove that a particular number is transcendental.
		The numbers pi and e can be expressed as an endless continued fraction or as the limit of an infinite series.
		If the number is terminating, convert it to non-terminating by subtracting one from the last digit, and adding an infinite string of 9's to the end.
	sprott.physics.wisc.edu /pickover/trans.html (1263 words)

	Pi Facts
		An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers.
		Euler adopted the symbol and it quickly became a standard notation.
		The sum of the numbers in every column is duplicated by a sum of numbers in every row.
	www.geocities.com /SiliconValley/Pines/5945/facts.html (2032 words)

	LEONHARD EULER (Site not responding. Last check: 2007-09-07)
		Euler's contribution to mathematics is represented here by a few of the notations conventionalized by him or in his honor.
		Euler grew up near Basel, Switzerland, and studied at an early age under Johann Bernoulli.
		Leonhard Euler 1707-1783, Beiträge zu Leben und Werk, Gedenkband des Kantons Basel-Stadt, edited by J. Burckhardt, E. Fellmann, and W. Habicht, Birkhäuser Verlag, Basel, 1983.
	faculty.evansville.edu /ck6/bstud/euler.html (331 words)

	Euler number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-07)
		Euler number -- Facts, Info, and Encyclopedia article
		The Euler numbers are a (Serial arrangement in which things follow in logical order or a recurrent pattern) sequence E
		The (Click link for more info and facts about Euler polynomials) Euler polynomials are constructed with the Euler numbers.
	www.absoluteastronomy.com /encyclopedia/e/eu/euler_number.htm (159 words)

	Problem L - GWCF Contest 2 - Joint Effort Contest
		Memory Limit: 40 MB “That every number which is resolvable into two prime numbers can be resolved into as many prime numbers as you like, can be illustrated and confirmed by an observation which you have formerly communicated to me, namely that every even number is a sum of two primes, and since
		However, that every number is a sum of two primes, I consider a theorem which is quite true, although I cannot demonstrate it.”
		Your task is to test the conjecture for specified integers, considering that prime numbers are the numbers which are positive and divisible by exactly two positive integers.
	acm.uva.es /p/v103/10311.html (273 words)

	Euler Numbers Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-09-07)
		Looking For euler numbers - Find euler numbers and more at Lycos Search.
		Find euler numbers - Your relevant result is a click away!
		Look for euler numbers - Find euler numbers at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Euler_numbers (309 words)

	Euler Numbers and Euler Polynomials
		Compute the Euler number or the Euler polynomial.
		Note that Euler numbers are only defined for 0 and positive even integers.
		The Euler polynomials can be defined in terms of the Euler numbers:
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/en.htm (216 words)

	Euler's Sequence (Site not responding. Last check: 2007-09-07)
		Its limit, by definition, will be called e for Euler's number.
		Euler's number e is irrational (in fact transcendental), and an approximation of e to 30 decimals is e ~ 2.71828182845904523536028747135.
		Hence, Euler's sequence is bounded by 3 for all n.
	pirate.shu.edu /projects/reals/numseq/s_euler.html (187 words)

	Continued Fractions - An introduction
		Since the numbers always reduce, that is, the size of the remaining rectangle left over will always have one side smaller than the starting rectangle, then the process will always stop with a final n-by-1 rectangle.
		If we look at irrational numbers (numbers which cannot be written exactly as a fraction) we will find no pattern in their decimal fractions.
		The numbers that are squared are Fibonacci numbers (starting with 0 and 1 we add the latest two numbers to get the next, giving the series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...).
	www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/cfINTRO.html (8242 words)

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