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Topic: Eulers theorem

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Leonhard Euler

Euler characteristic

Euler angles

Euler Mascheroni constant

Euler equations

Euler Lagrange equations

Euler numbers

Euler Maclaurin formula

Euler product

Euler Jacobi pseudoprime

Leonard Euler

Euler pseudoprime

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	Euler
		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 became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.
		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.
	www-groups.dcs.st-and.ac.uk /history/Mathematicians/Euler.html (4240 words)

	Gresham College \| Search Lectures & Events (Site not responding. Last check: 2007-10-11)
		Euler, ‘the Mozart of mathematics’, was probably the most prolific mathematician of all time, having contributed to many areas, both theoretical and practical, both serious and recreational – yet he remains largely unknown except to mathematicians.
		Euler worked in an astonishing variety of areas, ranging from the very pure the theory of numbers, the geometry of a circle and musical harmony via such areas as infinite series, logarithms, the calculus and mechanics, to the practical optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides.
		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.
	www.gresham.ac.uk /event.asp?PageId=4&EventId=67 (3622 words)

	Number theory - Wikipedia
		Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity.
		The properties of number theoretical functions such as the Möbius function and Euler's φ function are investigated; so are integer sequences such as factorials and Fibonacci numbers.
		The prime number theorem and the related Riemann hypothesis are examples.
	nostalgia.wikipedia.org /wiki/Number_theory (545 words)

	Euler
		Eulers first major breakthrough came when he won second place in the "Grand prize of the Paris Academy" in 1727, and since then he never looked back, proving major result after major result.
		Eulers passion for his subject never waned, it is often said he would attempt to dash of a paper inbetween the first and second calls for dinner, and if he was craddling an infant in one hand, his other would be finishing off a detailed proof or calculation.
		Eulers extreme mathematical ability is shown by the fact that he continued with mathematics after he lost his sight, in fact his level of production increased.
	people.bath.ac.uk /pb242 (395 words)

	Euler's theorem - Open Encyclopedia (Site not responding. Last check: 2007-10-11)
		In number theory, Euler's theorem (also known as the Fermat-Euler theorem) states that if n is a positive integer and a is relatively prime to n, then
		The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.
		Using modern terminology, one may prove the theorem as follows: the numbers a which are relatively prime to n form a group under multiplication mod n, the group of units of the ring Z/nZ.
	www.open-encyclopedia.com /Euler%27s_theorem (224 words)

	Encyclopedia: Augustin Louis Cauchy
		He started the project of formulating and proving the theorems of calculus in a rigorous manner and was thus an early pioneer of analysis.
		His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions, differential equations and determinants.
		He clarified the principles of the calculus by developing them with the aid of limits and continuity, and was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.
	www.nationmaster.com /encyclopedia/Augustin-Louis-Cauchy (2534 words)

	Encyclopedia: Chinese remainder theorem
		The Chinese remainder theorem is the name for several related results in abstract algebra and number theory.
		The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory.
		Riemann (1859) conjectured the limit of the number of primes not exceeding a given number (the prime number theorem), introduced complex analysis into the theoryof the Riemann zeta function, and derived the explicit formulae of prime number theory from its zeroes.
	www.nationmaster.com /encyclopedia/Chinese-remainder-theorem (266 words)

	Planar graph (Site not responding. Last check: 2007-10-11)
		Therefore, by Theorem 2, it is not planar.
		Eulers formula can be proven as follows: if the graph isnt a tree (graph theory), then remove an edge which completes a cycle.
		A theorem of Steinitz says that the planar graphs formed from convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite graph theory simple planar graphs.
	read-and-go.hopto.org /Graphs/Planar-graph.html (1039 words)

	Encyclopedia: Pierre de Fermat
		Fermat is famous for his "Enigma" that was an extension of Pythagorean Theorem, also known as Fermat's last theorem, which baffled mathematicians for more than 300 years, and was only finally proven in 1994.
		Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you take some number a, multiply it by itself p times and subtract a, the result is divisible by p...
		Fermat was for some time councillor for the parliament of Toulouse, and in the discharge of the duties of that office he was distinguished both for legal knowledge and for strict integrity of conduct.
	www.nationmaster.com /encyclopedia/Pierre-de-Fermat (1825 words)

	References for Euler
		C Grau, Leonhard Euler und die Berliner Akademie der Wissenschaften, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 139-149.
		H-J Treder, Euler und die Gravitationstheorie, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 112-119.
		A P Yushkevich, Leonhard Euler - sein Leben und mathematisches Werk, in Ceremony and scientific conference on the occasion of the 200th anniversary of the death of Leonhard Euler (Berlin, 1985), 17-33.
	www-groups.dcs.st-and.ac.uk /history/References/Euler.html (1152 words)

	LEONHARD EULER (Site not responding. Last check: 2007-10-11)
		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's totient function : Eulers phi function
		Euler's totient function φ(n), named after Leonhard Euler, is an important function in number theory.
		If n is a positive integer, then φ(n) is defined to be the number of positive integers less than or equal to n and coprime to n.
		This, together with Lagrange's theorem, provides a proof for Euler's theorem.
	www.fastload.org /eu/Eulers_phi_function.html (260 words)

	De Moivre's formula - Wikipedia, the free encyclopedia
		Abraham de Moivre was a good friend of Newton; in 1698 he wrote that the formula had been known to Newton as early as 1676.
		It can be derived from (but historically preceded) Euler's formula e
		Hence, the theorem is true for all integral values of n.
	en.wikipedia.org /wiki/De_Moivres_formula (370 words)

	ENERGETICS - LoveToKnow Article on ENERGETICS
		Historical: Abstract Dynamics.Even in the case of a purely mechanical system, capable only of a finite number of definite types of disturbance, the principle of the conservation of energy is very far from giving a complete account of its motions; it forms only one among the equations that are required to determine their course.
		In its application to the kinetics of invariable systems, after the time of Newton, the principle was emphasized as fundamental by Leibnitz, was then improved and generalized by the Bernoullis and by Euler, and was ultimately expressed in its widest form by Lagrange.
		Increasing the volume 4 times, and all the masses to the same extentin fact, placing alongside each other 4 identical systems at the same temperature and pressurewill increase~andE in the same ratio 4; thus E must be a homogeneous function of the first degree of the independent variables 4, 1, in i, -.
	49.1911encyclopedia.org /E/EN/ENERGETICS.htm (7891 words)

	Euler's Formula
		Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways.
		Examples of this include the existence of infinitely many prime numbers, the evaluation of zeta(2), the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a formula for testing whether an arithmetic progression contains a square) and the Pythagorean theorem (which according to Wells has at least 367 proofs).
		Perhaps there is a proof of Euler's formula that uses these polynomials directly rather than merely translating one of the inductions into polynomial form.
	www.ics.uci.edu /~eppstein/junkyard/euler (615 words)

	Talk:The most remarkable formula in the world - TheBestLinks.com - Complex number, De Moivres formula, Eulers number, ...
		The most remarkable formula in the world is an example of Euler's Theorem from Complex Analysis, which states that
		The proof of Euler's Theorem involves the definition of e, by a Taylor's series expansion of e
		Despite this last remark, Euler's Theorem is considered a direct consequence of the extension of the definition of the function e
	www.thebestlinks.com /Talk__3A__The_most_remarkable_formula_in_the_world.html (1014 words)

	Amazon.co.uk: Gamma: Exploring Euler's Constant: Books (Site not responding. Last check: 2007-10-11)
		Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!).
		Havil has filled the gap by writing a book about Euler's constant, Gamma; it was the most obvious number missing from the growing library of 'popular' mathematical explanations.
		The historical approach of these expositions has great appeal and provides welcome respite from some of the actual mathematics; in the case of this book, there really is a great deal of mathematics too.
	www.amazon.co.uk /exec/obidos/ASIN/0691099839 (1282 words)

	Euler's totient function at opensource encyclopedia (Site not responding. Last check: 2007-10-11)
		Euler's totient function φ(n) is defined, for any positive integer n, to be the number of positive integers less than or equal to n and coprime to n.
		Named after the Swiss mathematician Leonhard Euler, φ(n) is an important function in number theory, chiefly because it is the cardinality of the multiplicative group of integers modulo n — more precisely, the order of the group of units of the ring Z/nZ (see modular arithmetic).
		This fact, together with Lagrange's theorem, provides a proof for Euler's theorem.
	wiki.tatet.com /Euler%27s_totient_function.html (508 words)

	Math Forum: Leonard Euler and the Bridges of Konigsberg
		A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.
		The 'Euler number' of a 'network' like the ones presented later in this discussion is an example of a property that does not change when the network is stretched or compressed.
		One of these areas is the topology of networks, first developed by Leonard Euler in 1735.
	mathforum.org /isaac/problems/bridges1.html (321 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		In the study of numbers, one of the most important ideas is called Euler's Totient Function, denoted as ET(I).
		Euler recognized that a very important number associated with any integer, I, is the number of integers less than I which share no common factors with I. Here is a short list:
		Stumbling upon theorems like this, which apparently have no connection to the Farey sequence is one of the things makes math fun for us..
	www.ridgecrest.ca.us /~jebush/bcf/eul/Sent3tot.html (906 words)

	Theorem - Definition of Theorem by Webster's Online Dictionary (Site not responding. Last check: 2007-10-11)
		That which is considered and established as a principle; hence, sometimes, a rule.
		Not theories, but theorems (), the intelligible products of contemplation, intellectual objects in the mind, and of and for the mind exclusively.
		a theorem which expresses the impossibility of any assertion.
	www.webster-dictionary.org /definition/theorem (109 words)

	Bernoulli Bibliography: S (Site not responding. Last check: 2007-10-11)
		[1] An extension of the Bernoulli and Euler polynomials, Jandntildeanabha, 20 (1990), 7-12.
		[1] On unification and extension of Bernoulli, Euler and Eulerian polynomials.
		[1] On the extension of Bernoulli, Euler and Eulerian polynomials.
	www.mathstat.dal.ca /~dilcher/berns.html (4626 words)

	Arkivets særtryksamling (Site not responding. Last check: 2007-10-11)
		Bohr, Harald: A survey of the different proofs of the main theorems in the theory of almost periodic functions.
		Bohr, Harald & Jessen, Børge: Mean-value theorems for the Riemann zeta-function.
		Jessen, Børge: The algebra of polyhedra and the Dehn-Sydler theorem.
	www.math.ku.dk /arkivet/offprint/reprs.htm (6138 words)

	MAS231, Geometry II: Knots and Surfaces
		Gauss-Bonnet theorem for integral of geodesic curvature in terms of integral of Gauss curvature in the interior, for simple closed curves and for curvilinear n-gons.
		The student should be able to state some of the main theorems and be able to reproduce some of the shorter proofs or parts of proofs.
		The student should also be able to demonstrate understanding of the main theorems through examples.
	www.maths.qmw.ac.uk /undergraduate/modules/MAS231.html (388 words)

	References for Euler
		R Ayoub, Euler and the zeta function, Amer.
		E A Fellmann, Leonhard Euler 1707-1783 : Schlaglichter auf sein Leben und Werk, Helv.
		E Knobloch, Eulers früheste Studie zum Dreikörperproblem, Amphora (Basel, 1992), 389-405.
	www-history.mcs.st-andrews.ac.uk /history/References/Euler.html (1152 words)

	Number Theory (Site not responding. Last check: 2007-10-11)
		Summary of Eulers 96 Articles on Number Theory - Part of the Euler Project.
		Fermat's Little Theorem - With notes on Carmichael numbers and the life of R.D. Carmichael.
		Infinite Series Theorem - Addresses the question whether a series of rational functions converges to a rational number.
	www.topiasearch.com /Science/Math/NumberTheory (641 words)

	Euler : The Master of Us All (Site not responding. Last check: 2007-10-11)
		Eulers works are not readily available and this book is one rare chance to see some fully worked proofs of Eulers.
		Luckily for me my university has Eulers "Opera Omnia" in its library, but even for us privaleged Euler fans, the great tomes are in all sorts of languages, (Latin, German, French andc).
		It explores the deeds of Euler in the different fields of math, one chapter to a field, and gives the most basic insights and results.
	www.bookhub.co.uk /Euler__The_Master_of_Us_All_jhnza_0883853280.html (455 words)

	Citations: Eulers Charakteristik und kombinatorische Geometrie - Hadwiger (ResearchIndex) (Site not responding. Last check: 2007-10-11)
		Note that theorem 14 gives an inductive definition of the Euler Poincar e number in terms of sections of lower dimensions.
		It was applied to volume computation by Bieri Nef [5] and (with a different name) by Lawrence [22] see also [14] In [4] a recursive algorithm is used to count the cells of a finite division in R n.
		, Sa] In category 2 we have the Jordan separation theorem, The Borsuk Ulam theorem, the Poincare Hopf index theorem of Topology; Rouche s theorem and the Gauss Lucas in complex variables; the fundamental theorem of algebra and the intermediate value theorem of elementary Mathematics; and the not.
	citeseer.ist.psu.edu /context/233544/0 (755 words)

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