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Topic: Fermat curve

	PlanetMath: Fermat's last theorem
		Fermat's last theorem was actually a conjecture and remained unproved for over 300 years.
		We cannot imagine how Fermat's last theorem could be proved without these advanced mathematical tools, which include group theory and Galois theory, the theory of modular forms, Riemannian topology, and the theory of elliptic equations.
		Assuming Fermat's teaser was truthful, and Fermat was not in error, this apparent paradox has led some to jokingly attribute supernatural abilities to Fermat.
	planetmath.org /encyclopedia/FermatsLastTheorem.html (582 words)

	ELLIPTIC CURVES IN NATURE (Site not responding. Last check: 2007-10-20)
		This curve E is a quotient of the Jacobian of J_0(98) -- indeed so is its restriction of scalars to Q (which is isogenous to E^2 because E is isogenous with its Galois conjugate), which corresponds to a pair of Hecke eigenforms of level 98 with coefficients in Z[sqrt(2)].
		The curve 30-A2(B) is likewise the Frey curve for 5+27=32.
		Fermat proved that these curves have no rational points other than their torsion points, using what he famously called his "method of descent", and which is a special case of what we now call a descent via a 2-isogeny, here the 2-isogeny between 32-A1(B) and 32-A2(A).
	modular.fas.harvard.edu /Tables/nature (5086 words)

	Pierre de Fermat
		Pierre de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small town near Toulouse in the south part of France, near the border with Spain.
		Fermat's choice of a legal career was natural and typical of his time, for his father's wealth and his mother's famil y background.
		"From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence(in refraction)is a constant number times the sine of the angle of refraction in passing from one medium to anot her." (Bell, p.63).
	www.math.rutgers.edu /~cherlin/History/Papers1999/chellani.html (1446 words)

	Timeline of Fermat's Last Theorem
		Fermat published anonymously a dissertation on the rectification of curves Lalouvé published in 1660 as an appendix to his book on the cycloid.
		Fermat pondered publication of his work on a few occasions, but he insisted on anonymity because the amount of supervision required to produce an adequate copy.
		Since the genus of the Fermat equation for n > 3 was 2 or more, it became evident that the integer solutions to the Fermat equation were finite, if they existed at all.
	www.public.iastate.edu /~kchoi/time.htm (2119 words)

	Fermat curve - Wikipedia, the free encyclopedia
		In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation
		This curve therefore can be used to formulate Fermat's last theorem in a geometric way (hence its name).
		The Jacobian variety of the Fermat curves has been studied in depth.
	en.wikipedia.org /wiki/Fermat_curve (167 words)

	History of the Differential from the 17th Century
		Some credit Fermat with discovering the differential, but it was not until Leibniz and Newton rigorously defined their method of tangents that a generalized technique became accepted.
		Fermat's first documented problem in differentiation involved finding the maxima of an equation, and it is clearly this work that led to his technique for finding tangents.
		Fermat again lets the quantity E = 0 (in modern term, he took the limit as E approached 0) and recognized that the bottom portion of the equation was identical to his differential in his method of mimina.
	www.math.wpi.edu /IQP/BVCalcHist/calc2.html (1805 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		By profession Fermat was a civil servant and judge, responsible for dealing with some of the most serious cases, including the condemnation of priests to be burned at the stake.
		According to Fermat, none of these equations could be solved, and he noted this in the margin of his Arithmetica.
		Fermat believed he could prove his theorem, but he never committed his proof to paper.
	www.prometheus.demon.co.uk /01/01fermat.htm (4279 words)

	The Whole Story (Site not responding. Last check: 2007-10-20)
		The history of Fermat’s Last Theorem is a tale of intrigue, rivalry, rich prizes, suicide and death, involving characters who became obsessed by Fermat’s accidental challenge.
		Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture.
		Wiles’ proof of Fermat’s Last Theorem relies on verifying a conjecture born in the 1950s, which in turn shows that there is a fundamental relationship between elliptic curves and modular forms.
	www.simonsingh.net /FLT_the_whole_story.html (3998 words)

	Infinitesimals and Transcendent Relations
		The curves he was talking about simply did not belong to classical mathematics either in its original geometrical form or in the extended algebraic form given it by Descartes.
		That physical mechanism became the geometrical configuration of a curve which was "unwound" by drawing successive tangents to it and laying off on them segments equal to the arc length of the curve from some fixed origin to the point of tangency.
		That the straight and the curved coalesced at the level of the infinitesimal was a premiss of the calculus, indeed its raison d'être.
	www.princeton.edu /~mike/articles/canons/canons.htm (8958 words)

	The cuspidal torsion packet on the Fermat curve, by Robert F. Coleman, Akio Tamagawa and Pavlos Tzermias (Site not responding. Last check: 2007-10-20)
		The cuspidal torsion packet on the Fermat curve, by Robert F. Coleman, Akio Tamagawa and Pavlos Tzermias
		The cuspidal torsion packet on the curve is defined as the set of points on the curve whose image under this embedding is a torsion point on the Jacobian.
		In this paper, we prove that the cuspidal torsion packet on such a curve is the set of cusps.
	www.math.uiuc.edu /Algebraic-Number-Theory/0062 (151 words)

	Horizon - Fermat's Last Theorem
		Fermat owned a copy of this book, which is a book about numbers with lots of problems, which presumably Fermat tried to solve.
		Fermat's original notes were lost, but they can still be read in a book published by his son.
		Well I was at this conference on L functions and elliptic curves and it was kind of a standard conference and all of the people were there, didn't seem to be anything out of the ordinary, until people started telling me that they'd been hearing weird rumours about Andrew Wiles's proposed series of lectures.
	www.cs.wichita.edu /~chang/fermat.html (4959 words)

	Fermat, Pierre de
		The curves determined by this equation are known as the parabolas or hyperbolas of Fermat according as n is positive or negative (Kolata).
		Fermat was out of touch with his scientific colleagues in Paris during the period from 1643 to 1654 due to many complications.
		Fermat described his method of infinite descent, also known as "reverse induction" (Mahoney, page 231), and gave an example on how it could be used to prove that ever y prime of the form 4k+1 could be written as the sum of two squares.
	www.math.rutgers.edu /courses/436/436-s00/Papers2000/pellegrino.html (3192 words)

	PhysOrgForum Science, Physics and Technology Discussion Forums -> Fermat
		Fermat's n=4 proof does not test all possible integer solutions of of X, Y and Z; consequently, Fermat's n=4 proof is invalid.
		Fermat's elliptical curve proof also bounds the solutions to an elliptical curve equation; therefore, Fermat's elliptical curve method is also invalid since the elliptical curves do not represent all possible integer solutions.
		Fermat's n=4 and elliptical curve proofs are invalid since both methods bound the solutions.
	forum.physorg.com /index.php?showtopic=5776 (416 words)

	UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem
		Let X denote the modular curve whose points correspond to pairs (A, C) where A is an elliptic curve and C is a subgroup of A isomorphic to the group scheme E[5].
		Fermat claimed to have found a proof of the theorem at an early stage in his career.
		Only on one ill-fated occasion did Fermat ever mention a curve of higher genus x^n + y^n = z^n, and then hardly remain any doubt that this was due to some misapprehension on his part [for a brief moment perhaps [he must have deluded himself into thinking he had the principle of a general proof.
	www.math.uncc.edu /flt.php (3199 words)

	Fermat's Last Theorem - Wikipedia, the free encyclopedia
		Fermat's Last Theorem is one of the most famous theorems in the history of mathematics.
		The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi.
		Fermat's Last Theorem is a generalisation of this result to higher powers n, and states that no such solution exists when the exponent 2 is replaced by a larger integer.
	en.wikipedia.org /wiki/Fermat's_last_theorem (2156 words)

	Fermat's spiral (Site not responding. Last check: 2007-10-20)
		The spiral of Fermat is a kind of Archimedean spiral.
		It was the great mathematician Fermat (1636) who started investigating the curve, so that the curve has been given his name.
		Sometimes the curve is called the dual Fermat's spiral when both both negative and positive values are accepted.
	www.2dcurves.com /spiral/spiralf.html (101 words)

	The Parabola of Fermat - National Curve Bank: A MATH Archive
		This parabolic curve was investigated by Fermat as early as 1636.
		The spiral curves are easily entered and modified on a graphing calculator.
		The name spiral, where a curve winds outward from a fixed point, has been extended to curves where the tracing point moves alternately toward and away from the pole, the so-called sinusoidal type.
	curvebank.calstatela.edu /parabolafermat/parabolafermat.htm (600 words)

	Math Trek: Fermat's Natural Spirals, Science News Online, Sept. 3, 2005
		One way to model such a pattern is to start with a curve called Fermat's spiral.
		This curve is also known as a parabolic spiral.
		Fermat's spiral, in particular, "is a natural basis for this inward draw."
	www.sciencenews.org /articles/20050903/mathtrek.asp (649 words)

	The Mathematics of Fermat's Last Theorem
		All semistable elliptic curves with rational coefficients are modular.
		Elliptic curves are relatively simple objects that helped inspire the field of algebraic geometry because of some very special properties.
		There turn out to be many parallels between the theory of elliptic curves and that of modular functions, which have deep consequences for both theories.
	www.mbay.net /~cgd/flt/fltmain.htm (2364 words)

	Fermats (Site not responding. Last check: 2007-10-20)
		This spiral was discussed by Fermat in 1636.
		The inverse of Fermat's Spiral, when the pole is taken as the centre of inversion, is the spiral r
		For technical reasons with the plotting routines, when evolutes, involutes, inverses and pedals are drawn only one of the two branches of the spiral are drawn.
	www-groups.dcs.st-and.ac.uk /~history/Curves/Fermats.html (117 words)

	A GELLMU Demonstration
		It should be noted that the fact that “Fermat's Last Theorem” is a consequence of sufficient knowledge of the theory of “elliptic curves” has been fully documented in the publications ([14], [15]) of K. Ribet.
		Elliptic curves are the “group objects” in the category of algebraic curves that reside in projective space: for each extension field K of k the set E(K) of “K-valued points” of E is an abelian group.
		Thus, the classification of elliptic curves over C does not lead directly to the desired enumerative classification of elliptic curves defined over Q, but it does bring to the fore the notion of modular form, which is central in the study of elliptic curves defined over Q.
	www.albany.edu /~hammond/gellmu/examples/f356g.html (6195 words)

	The Proof of Fermat's Last Theorem
		Call this curve E. Frey noted it had some very unusual properties, and guessed it might be so unusual it could not actually exist.
		Suppose E is a semistable elliptic curve with conductor N and that its associated Galois representation
		In the following, we assume that E is a semistable elliptic curve with conductor N. We have to prove E is modular.
	www.mbay.net /~cgd/flt/flt08.htm (1543 words)

	Read This: Invitation to the Mathematics of Fermat- Wiles
		There are some good books regarding the history of the problem: The Fermat Diary, Fermat's Enigma; and some which deal with the basic mathematics of FLT: Fermat's Last Theorem for Amateurs.
		, a proof of Mason's Theorem on the parametrizability of curves, the algebra of quadratic fields, and the geometry of lattices.
		This chapter has the longest set of exercises; much of it has to do with algebraic geometry, and the last problem develops many of the standard results about congruent numbers (a la Koblitz).
	www.maa.org /reviews/fermatwiles.html (1146 words)

	lituus (Site not responding. Last check: 2007-10-20)
		The lituus is a species of the Archimedean spiral.
		The curve is formed by the loci of a point P moving in such a way that the area of a circular sector remains constant.
		The inverse of the curve is Fermat's spiral.
	www.2dcurves.com /spiral/spirall.html (100 words)

	Generators of x^3+y^3=p for prime p=9k+4<5000 or 9k+7<10000
		For n=1, the curve E(n) is the Fermat cubic curve, known [Euler?] to have no rational points except (1:-1:0) and the 3-torsion points (-1:0:1) and (0:1:-1).
		For general n, the curve E(n) is a ``twisted Fermat cubic''.
		These are the cases in which the curve has odd analytic rank, and since the 3-descent yields an upper bound of 1, the arithmetic rank should be exactly 1.
	www.math.harvard.edu /~elkies/sel_p.html (807 words)

	Examples of curves
		A curve is an ``absolutely irreducible projective curve defined over a field field''.
		Such a curve is determined by a polynomial
		The Hermitian curve is a special case of the Fermat curve.
	web.usna.navy.mil /~wdj/book/node206.html (43 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		It is known that the points on this curve with coordinates in Q form a finitely generated group, but no algorithm is known for finding the rank of the group.
		A key to the proof of Fermat's Last Theorem was the realization that a point on a Fermat curve would give rise to an elliptic curve over Q with properties so weird that it couldn't possibly exist.
		A projective algebraic variety with a group structure is called an abelian variety---elliptic curves are the abelian varieties of dimension one.
	www.jmilne.org /math/Personal/profile.html (344 words)

	The Hyperbola of Fermat - National Curve Bank: A MATH Archive
		What is less well-known is that Fermat, not Descartes, might be credited with writing about these curves earlier than his contemporary.
		The following are all known as the hyperbola, parabola and spiral of Fermat.
		In a letter written to Roberval in 1636, Fermat stated that he had formulated these curves seven years earlier.
	curvebank.calstatela.edu /hyperbolafermat/hyperbolafermat.htm (695 words)

	[No title]
		Sine the curve has rank 0 over Q (by Fermat!) the rank over Q(sqrt(d)) is equal to the rank over Q of the twisted curve Y^2 = X^3 - 432 d^3.
		Perhaps surprisingly, there is an exact correspondence between rational points on this elliptic curve and points on the Fermat cubic with coordinates in Q(sqrt{d}).
		Then this line necessarily intersects the Fermat cubic in two further points (by Bezout's theorem, if we work in P^2(C)); it is easy to see that these lie in some quadratic field and are conjugate.
	www.math.niu.edu /~rusin/known-math/00_incoming/flt_rings (3038 words)

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