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Topic: Method of infinite descent

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	Infinite descent - Wikipedia, the free encyclopedia
		In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction.
		More precisely, his method of infinite descent was an exploitation in particular of the possibility of halving rational points on an elliptic curve E by inversion of the doubling formulae.
		In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it connected with the main thrust of algebraic number theory a == nd the study of L-functions.
	en.wikipedia.org /wiki/Infinite_descent (510 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		As used by Fermat, for FLT cases and many other problems, the method of descent starts with one (or more) equations, and then from these via arithmetic reasoning derives equation[s] identical in form to the first but for which the size of the unknowns is in some sense less than the first.
		The Method of Descent is most commonly used with homogenous equations, for which multiplying (or dividing) every unknown by the same factor leaves the equations unchanged.
		To use descent to prove the equation[s] impossible, one starts by assuming a solution with a non-zero height, and then one shows that in the derived system, with the smaller height, the height is _still_ non-zero.
	www.math.niu.edu /~rusin/known-math/99/descent (740 words)

	Infinity
		The dialectical puzzles of the fifth-century Eleatics, sharpened by Plato and Aristotle in the fourth century, are complemented by the invention of precise methods of limits, as applied by Eudoxus in the fourth century and Euclid and Archimedes in the third.
		The very suggestion that certain objects, infinite in number, are "equal in magnitude" to others implies that not all such objects, infinite in number, are so equal.
		the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes "equal", "greater", and "less" are not applicable to the infinite, but only to finite quantities.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Infinity.html (3916 words)

	Infinite Descent (Site not responding. Last check: 2007-11-03)
		In an "Infinite Descent" proof, you first assume that a number with a certain property is minimal, and then you find another number with the same property that's smaller than the first number.
		Just one iteration of this descent is enough to prove by contradiction that there is no number with that certain property, but the method is called "Infinite" Descent because it shows more than just a contradiction; it shows that there is no smallest number with the given property.
		That is, the fact that the descent is infinite affords you the luxury of not needing to assume minimality in the first place.
	mcraeclan.com /MathHelp/PythagInfiniteDescent.htm (159 words)

	Fermat's Last Theorem: Infinite Descent (Site not responding. Last check: 2007-11-03)
		He did provide one example of this method in his proof that the area of a right triangle cannot be equal to a square number.
		An elegant application of this proof is found in the case of FLT: n=4 where the proof rests on the method of infinite descent and the solution to Pythagorean Triples.
		In this case, infinite descent is impossible since it contradicts the Well Ordering Principle.
	fermatslasttheorem.blogspot.com /2005/05/infinite-descent.html (714 words)

	PlanetMath: infinite descent (Site not responding. Last check: 2007-11-03)
		with certain properties implies that there exists a smaller one with these properties, then there are infinitely many of these, which is impossible.
		See here for a discussion of infinite descent vs. induction.
		This is version 10 of infinite descent, born on 2004-02-02, modified 2004-02-13.
	planetmath.org /encyclopedia/InfiniteDescent.html (133 words)

	Fermat's Infinite Descent (Site not responding. Last check: 2007-11-03)
		Pierre de Fermat's method of infinite descent is beautifully illustrated by the proofs of the following two propositions in Number Theory.
		In a letter to Carcavi describing his methods he wrote "As ordinary methods, such as are found in books, are inadequate to proving such difficult propositions, I discovered at last a most singular method...which I called the infinite descent.
		To apply it to affirmative questions is much harder, so when I had to prove 'Every prime of the form 4n+1 is a sum of two squares" I found myself in a sorry plight (en belle peine).
	www.mathpages.com /home/kmath288.htm (535 words)

	New Page 1
		For occasional demonstrations of his theorems Fermat used a device that he called his method of "infinite descent," an inverted form of reasoning by recurrence or mathematical induction.
		His superb intellect, his unusual powers of observation, and his mastery of the art of drawing led him to the study of nature itself, which he pursued with method and penetrating logic--and in which his art and his science were equally revealed.
		Appolonius The two lives contained in the Laurentian manuscript of the Argonautica say that Apollonius was a pupil of Callimachus; that he gave a recitation of the Argonautica at Alexandria; and that when this proved a failure he retired to Rhodes.
	www.geocities.com /crazymathematics/mathematicians.htm (4355 words)

	[No title]
		] Any method of solving magnetostatic, hydrodynamic, and other problems involving boundary conditions at the interface between two media, in which fictitious objects, such as magnetic dipoles and sources and sinks of fluid, are introduced to satisfy the boundary conditions; these methods are generalizations of the method in electrostatics.
		] A method of determining the heat of fusion of a substance whose specific heat is known, in which a known amount of the solid is combined with a known amount of the liquid in a calorimeter, and the decrease in the liquid temperature during melting of the solid is measured.
		] A method of estimating the parameters of a frequency distribution by first computing as many moments of the distribution as there are parameters to be estimated and then using a function that relates the parameters to moments.
	www.accessscience.com /Dictionary/M/M22/DictM22.html (2140 words)

	Fermat
		However some of his methods were published, for example Hérigone added a supplement containing Fermat's methods of maxima and minima to his major work Cursus mathematicus.
		Descartes attacked Fermat's method of maxima, minima and tangents.
		Fermat described his method of infinite descent and gave an example on how it could be used to prove that every number of the form 4k+1 could be written as the sum of two squares.
	sfabel.tripod.com /mathematik/database/Fermat.html (2471 words)

	Method of Steepest Descent
		The method of Steepest Descent is the simplest of the gradient methods.
		This implementation of the Steepest Descent method are often referred to as the optimal gradient method.
		In other words, the Steepest Descent method can be used where one has an indication of where the minimum is, but is generally considered to be a poor choice for any optimization problem.
	trond.hjorteland.com /thesis/node26.html (568 words)

	ABC-Dir: Method
		The method includes alphabet basics plus all the phonics that Japanese children need to read English.
		Describes a method that is to be used for building varying interfaces on the basis of the users' needs.
		A method to conquer excessive anxiety and inhibitions.
	www.abc-directory.com /view/method (205 words)

	Thomas Aquinas [Internet Encyclopedia of Philosophy]
		This long association of Thomas with the great polyhistor was the most important influence in his development; it made him a comprehensive scholar and won him permanently for the Aristotelian method.
		From it follow an impairment and perversion of human nature in which thenceforth lower aims rule contrary to nature and release the lower element in man. Since sin is contrary to the divine order, it is guilt, and subject to punishment.
		Guilt and punishment correspond to each other; and since the "apostasy from the invariable good which is infinite," fulfilled by man, is unending, it merits everlasting punishment.
	www.iep.utm.edu /a/aquinas.htm (3032 words)

	Timeline of Fermat's Last Theorem
		Eudoxus was born in Cnidos, and became a colleague of Plato.
		He contributed to the theory of proportions, and invented the "method of exhaustion." This is the same method employed in integral calculus.
		He produced a method of determining when a general equation could be solved by radicals.
	www.public.iastate.edu /~kchoi/time.htm (2119 words)

	diophanfin.html
		Diophantus' methods of solving problems have had both lasting effects and great benefits for the studies of algebra and number theory.
		He developed his method of infinite descent as a method of proof.
		To inhabitants of such a world, the universe would appear to be infinite; and rays of light, or 'straight lines', would not be rectilinear, but would be circles orthogonal to the limiting sphere and would appear to be infinite." (Boyer 604).
	www.ms.uky.edu /~carl/ma330/projects/diophanfin1.html (2434 words)

	sci.math FAQ: Did Fermat prove FLT?
		* Fermat discovered and applied the method of infinite descent, which, in particular can be used to prove FLT for n = 4.
		This method can actually be used to prove a stronger statement than FLT for n = 4, viz, x^4 + y^4 = z^2 has no non-trivial integer solutions.
		It is possible and even likely that he had an incorrect proof of FLT using this method when he wrote the famous ``theorem''.
	www.faqs.org /faqs/sci-math-faq/FLT/Fermat (848 words)

	Diophantine Equations Science, Directory (Site not responding. Last check: 2007-11-03)
		Fermat's Method of Infinite Descent Notes by Jamie Bailey and Brian Oberg.
		Illustrates the method on FLT with exponent 4.
		There is also a link to his description of the solving methods.
	www.flashunion.org /ZmxzXzI2OTUw.aspx (461 words)

	Well-ordering principle - Wikipedia, the free encyclopedia
		This mode of argument bears the same relation to proof by mathematical induction that "If not B then not A" (the style of Modus tollens) bears to "If A then B" (the style of Modus ponens).
		It is known light-heartedly as the "minimal criminal" method and is similar in its nature to Fermat's method of "infinite descent".
		This page was last modified 16:52, 18 November 2005.
	en.wikipedia.org /wiki/Well-ordering_principle (201 words)

	Philosophers : Pierre de Fermat
		Some of his methods were published, for example Hérigone added a supplement containing Fermat's methods of maxima and minima to his major work Cursus mathematicus.
		He wrote to Fermat praising his work on determining the tangent to a cycloid (which is indeed correct), meanwhile writing to Mersenne claiming that it was incorrect and calling Fermat an inadequate mathematician and a thinker.
		He described his method of infinite descent and gave an example on how it could be used to prove that every number of the form 4k+1 could be written as the sum of two squares.
	www.trincoll.edu /depts/phil/philo/phils/fermat.html (1230 words)

	CATHOLIC ENCYCLOPEDIA: Aristotle
		For Aristotle, therefore, philosophic method implies the ascent from the study of particular phenomena to the knowledge of essences, while for Plato philosophic method means the descent from a knowledge of universal ideas to a contemplation of particular imitations of those ideas.
		In a certain sense, Aristotle's method is both inductive and deductive, while Plato's is essentially deductive.
		In the same treatise, he argues that, although motion is eternal, there cannot be an infinite series of movers and of things moved, that, therefore, there must be one, the first in the series, which is unmoved, to proton kinoun akineton--primum movens immobile.
	www.newadvent.org /cathen/01713a.htm (5735 words)

	Abstracts for Workshop Computational Arithmetic Geometry, July 5 - 9, 2004, Vancouver (Site not responding. Last check: 2007-11-03)
		Around 1930, Chevalley and Weil realized that the reduction step in Fermat's method of infinite descent could be applied whenever one had a finite unramified morphism of varieties X --> Y. Their idea has proved to be indispensable for the determination of the set of rational points on curves.
		In these cases, the Chevalley-Weil descent is useless, but the theory of Colliot-Thélène and Sansuc sometimes succeeds in determining the rational points.
		Investigating the obstruction is part of descent theory for fields of definition and has many consequences in arithmetic geometry.
	www.cecm.sfu.ca /~nbruin/WCAG2004/abstracts.html (959 words)

	International Society of TOE (Theory of Everything)
		Fermat himself proved the case of n=4 with the method of infinite descent.
		First, that the single point can be stretched into infinite points (as a circle) in Topology is viewed as a mathematical trick (as only convenience for mathematicians), not as an essence of mathematics.
		In fact, there are infinite amount of number which touches the number n from each side, but they can all be represented with a colored number (or a colored set).
	www.fortunecity.com /greenfield/crawdad/792/Fermat.htm (3844 words)

	UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem (Site not responding. Last check: 2007-11-03)
		For example, there is a classical proof that uses infinite descent to prove FLT for n = 4.
		The method used was essentially that of Wagstaff: enumerating and eliminating irregular primes by Bernoulli number computations.
		Fermat discovered and applied the method of infinite descent, which, in particular can be used to prove FLT for n = 4.
	www.math.uncc.edu /flt.php (3199 words)

	Rashed/ Islam and Science (Interview)
		But to go farther than that would have required the invention of a new method (the infinite descent) whose discovery was the privilege of Fermat.
		It is the combinations of such events, both external and internal to science, in the midst of economic and social decadence, as well as reaching the summit of a certain type of research, which are the causes behind the end of scientific innovation in Islamic civilization.
		In other words, it was at the moment when one had the direst need for intensifying the research at hand, which was already quite advanced, and when one needed to nurture it with other methods and other languages, that economic and social decadence made Islamic societies turn their back to science.
	cc.usu.edu /~bekir/rashed/interview_Islam_Science.htm (2963 words)

	[No title]
		It was Gottfried Leibniz, the 17th-century German mathematician and philosopher, and Leonhard Euler, the 18th-century Swiss mathematician, who took the trouble to prove this assertion.
		Fermat liked to do demonstrations of his theorems using a device that he called his method of "infinite descent," an inverted form of reasoning by recurrence or mathematical induction.
		Watch for publication of the series entitle The Imitation Game, in which an "infinite regress" is revealed in a famous mid-twentieth-century gedanken (ahem) formulated by Alan Turing.
	www.niquette.com /puzzles/xy-yxp.htm (533 words)

	monsur dot org (Site not responding. Last check: 2007-11-03)
		Though I devised a method that seemed to work, I was unable to prove it (I spent the better half of the semester trying).
		The method I pursued is vaguely similar to the method labeled "Approach by Induction of Inequalities"; the paper quickly loses me in mathematical jargon, so I'm not sure where the error actually occurs...
		The irony of all this is that Andrew Wiles actually used induction in order to prove that any given elliptic curve is also modular, which eventually lead to a proof of Fermat's Last Theorem.
	www.monsur.org /Main20020101_archive.html (1388 words)

	About "Fermat and His Method of Infinite Descent" (Site not responding. Last check: 2007-11-03)
		The basic method of the infinite descent is as follows: Assume one wants to prove no solution exists with a certain property.
		Repeat this argument an infinite number of times, thus infinitely descending through all integers.
		This contradicts the fact that there must be a smallest positive integer with this property.
	mathforum.org /library/view/17554.html (105 words)

	The method of descent
		In the language of the ladder metaphor, if you know you can't reach any rung without first reaching a lower rung, and you also know you can't reach the bottom rung, then you can't reach any rungs.
		The above is often called finite descent, to distinguish it from the variant method known as infinite descent:
		The classic example of infinite descent is in Fermat's proof that there are no positive integer solutions to
	mathcircle.berkeley.edu /BMC4/Handouts/induct/node7.html (264 words)

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