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Topic: Thue equations

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Diophantine equation

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	11D: Diophantine equations (Site not responding. Last check: 2007-11-06)
		Equations whose solutions are curves of genus 1 are discussed in the subsection on elliptic curves.
		Examples include cubics in two variables, homogeneous cubics in three variables, pairs of quadratics in four variables, and equations of the form y^2=Q(x) where Q is a polynomial of degree 3 or 4.
		Sets of N equations in N+2 variables (or N+3 variables, if those equations are homogeneous) describe algebraic surfaces ; for example the question of the existence of a "rational box" is there.
	www.math.niu.edu /~rusin/papers/known-math/index/11DXX.html (801 words)

	Diophantine equation - Wikipedia, the free encyclopedia
		In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers.
		A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.
		The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.
	www.wikipedia.org /wiki/Diophantine_equation (420 words)

	Solving equations
		Given a Thue object t and integers a, b, return the evaluation of the homogeneous polynomial f involved in t at (a, b), that is f(a, b).
		Given a Thue object t and an integer a this function return a sequence consisting of all sequences of two integers [x, y] which solve the equation f(x, y) = a, where f is the (homogeneous form of) the Thue equation associated with t.
		Given a Thue object t and an integer a this function return a sequence consisting of sequences of two integers [x, y] together providing all solutions to the equation f(X, Y)=a, where f is the (homogeneous form of) the polynomial associated with t.
	www.math.wisc.edu /help/magma/text454.html (515 words)

	Solving equations (Site not responding. Last check: 2007-11-06)
		Thue equations are Diophantine equations of the form f(x, y) = k, where k is some (integer) constant and f is a homogeneous polynomial in two variables.
		Given a polynomial of degree at least 2 over the integers, this function returns the `Thue object' corresponding to f; such objects are used by the functions for solving Thue equations.
		Given a Thue object t and integers a, b, this function returns the evaluation of the homogeneous polynomial f involved in t at (a, b), that is f(a, b).
	www.dtr.isy.liu.se /Magma/text446.html (520 words)

	Sharpening in the binomial case and an application
		Thue [36], [37], [38], [39] was the first to deduce classical Padé approximations for the binomial function and to apply it to the solution of (4).
		Thue's ineffective improvement of the theorem of Liouville made it possible for him to restrict the number of solutions of (4) in certain cases.
		Our method in paper III for the solution of (4) is essentially that of Thue refined in certain respects, especially by the use the common factor of the approximation polynomials as proposed by Chudnovsky [12].
	herkules.oulu.fi /isbn9514247191/html/A290/node5.html (1047 words)

	Solving Equations (Site not responding. Last check: 2007-11-06)
		Next we solve the same equation but come from a different angle, we will define the norm map as an element of the group ring and, instead of explicitly computing a relative extension, work instead with the implicit fix-field.
		To solve Thue equations the reduction of Bilu and Hanrot ( [BH96]) is used.
		Unit equations are equations of the form aepsilon + beta = c where a, b and c are some algebraic numbers and epsilon and eta are unknown units in the same field.
	magma.maths.usyd.edu.au /magma/htmlhelp/text640.htm (1952 words)

	Diophantine Equations - The Science Beat ... Scientific Research, Space, NASA - SearchBeat.com (Site not responding. Last check: 2007-11-06)
		Thue Equations - Definition of the problem and a list of special cases that have been solved.
		Hilbert's Tenth Problem - Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
		Quadratic Diophantine Equation Solver - Dario Alpern's Java/JavaScript code that solves Diophantine equations of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 in two selectable modes: "solution only" and "step by step" (or "teach") mode.
	www.searchbeat.com /Science/Math/NumberTheory/DiophantineEquations (519 words)

	Diophantine Equations Science, Directory (Site not responding. Last check: 2007-11-06)
		Solving General Pell Equations John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N. Sets of numbers such that the product of any two is one less than a square.
		Hilbert's Tenth Problem Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
		Thue Equations Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
	www.wacofdn.org /d2RjXzI2OTUw.aspx (472 words)

	Thue (Site not responding. Last check: 2007-11-06)
		Axel Thue was the son of Niels Thue and Nicoline Cathinka Eger.
		Then Thue was appointed to Trondheim Technical College where he worked from 1894 until 1903.
		In fact Thue wrote 35 papers on number theory, mostly on the theory of Diophantine equations, and these are reproduced in [2].
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Thue.html (401 words)

	Hilbert's Tenth Problem. Diophantine Equations. By K.Podnieks
		The equation (1) always represents in the (x, y)-plane a curve (an ellipse, a hyperbola, or a parabola), one or two straight lines, one isolated point, or nothing.
		Since D and a are not 0, this means that a solution (X, Y) of the reduced equation (2) yields a solution (x, y) of the equation (1), iff X-bd+2ae is divisible by D and Y-by-d is divisible by 2a (else x and y would not be integer numbers).
		Reduction of an arbitrary Diophantine equation to one in 13 unknowns.
	www.ltn.lv /~podnieks/gt4.html (4242 words)

	Clemens Heuberger - Thue equations (Site not responding. Last check: 2007-11-06)
		So the research interest in diophantine equations is to find classes of such equations which can be solved.
		This type of equations is called after him since then; he proved that such an equation only has a finite number of solutions.
		In 1990, E. Thomas studied a parametrized family of cubic Thue equations: It turns out that there exist only a few "trivial" solutions for large values of the parameter.
	finanz.math.tu-graz.ac.at /~cheub/thue.html (278 words)

	SAPO - Portugal Online!
		He is best known for his work to solve indeterminate equations in rational numbers.
		One of the puzzles he set was to find sets of unequal fractions such that the product of any two is one less than a square.
		Lower bounds for solving linear diophantine equations on random access machines.
	mundial.sapo.pt /Science/Math/Number_Theory/Diophantine_Equations (400 words)

	Tzanakis Publications (Site not responding. Last check: 2007-11-06)
		On the diophantine equation y2 - D = 2k, J.Number Th.17 (1983), 144-164.
		The diophantine equation x3 - 3xy2 - y3 = 1 and related equations, J.Number Th.
		Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms.
	www.math.uoc.gr /~www-old/2000-01/publ/tzanakis-publ.html (350 words)

	Clearing up the market cycle... best Thue Constant (Site not responding. Last check: 2007-11-06)
		If Periodia value is 26, that means that from the vary last maximum of its passed 26 ticks of time and we have "period length"/2-26 ticks of time to reverse point.
		Prouhet- Thue -Morse constant Prouhet- Thue -Morse constant The Prouhet- Thue -Morse constant is the number tau whose binary expansion is the Prouhet- Thue -Morse sequence.
		Thu e Constant -- from MathWorld Thue Constant -- from MathWorld The base-2 transcendental number 0.numeric20\dots_2 (Sloane's A014578), where the nth bit is 1 if n is not divisible by 3 and is the complement of the (n/3)th bit if n is...
	ascot.pl /th/Fourier5/Thue-Constant.htm (449 words)

	The Subspace Theorem
		There is a further generalization of this result, which we shall not state, dealing with systems of inequalities to be solved in vectors consisting of integers from a given algebraic number field.
		This generalization has a wide range of applications, such as finiteness results for Diophantine equations of the type considered in the previous sections, finiteness results for all sorts of Diophantine inequalities, transcendence results, finiteness results for integral points on surfaces, etc.
		Hence equation (10) may be viewed as a special case of (2).
	www.stieltjes.org /archief/rep20002001/node8.html (736 words)

	Publications
		A parametric family of quintic Thue equations II, Monatsh.
		A parametric family of quintic Thue equations, Math.
		Thue Equations associated with Ankeny- Brauer-Chowla Number Fields, J. London Math.
	www.uni-graz.at /~lettl/Literatur.html (454 words)

	ScienceDaily -- Browse Topics: Science/Math/Number_Theory/Diophantine_Equations (Site not responding. Last check: 2007-11-06)
		Thue Equations - Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
		Solving General Pell Equations - John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N. Pell's Equation - Record solutions.
		MAGMA program - MAGMA code to solve Diophantine equations of the form F(x)=G(y), for which Runge's condition is satisfied.
	www.sciencedaily.com /directory/Science/Math/Number_Theory/Diophantine_Equations (1101 words)

	References 2004-2005 (Site not responding. Last check: 2007-11-06)
		Tengely, "On the Diophantine equations Equation $x^2+a^2 = 2y^p$", 2000, p1-14.
		Clements Heuberger, "On a Conjecture of E. Thomas concerning parametrized Thue Equations", 2000, Aug. 3, p1-25.
		Clemens Heuberger, "On explicit bounds for the solutions of a class of parameterized Thue equations of arbitrary degree", p1-12.
	www.kaynet.or.jp /~kay/misc/refs.html (2620 words)

	On elliptic Diophantine equations that defy Thue's method: the case of the Ochoa curve, Roel J. Stroeker, Benjamin ...
		The purpose of this paper is to show that elliptic diophantine equations cannot always be solved--in the most practical sense--by the Thue approach, that is, by solving each of the finitely many corresponding Thue equations of degree 4.
		After a brief general discussion, which is necessarily of a heuristic nature, to substantiate our claim, we consider the elliptic equation associated with the Ochoa curve.
		An explicit computational explanation as to the reasons for the failure of the Thue approach in this case is followed by a complete solution of the standard Weierstraß equation of this elliptic curve by a method which makes use of a recent lower bound for linear forms in elliptic logarithms.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1048515872 (157 words)

	Publications of the SPACES team
		Given a geometric object defined by rational parametric equations, we show how to compute a disjunction of implicit equations and inequations that define exactly the same object by means of regular systems.
		We examine a system of equations arising in biophysics whose solutions are believed to represent the stable positions of N conical proteins embedded in a cell membrane.
		It is shown that these methods may be combined for automatically studying the number of admissible real solutions of a polynomial system depending on parameters; this study provides as output a description of this number as a function of the parameters.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13157 words)

	[No title]
		For instance, Thue's Theorem is used to give a short proof of Lagrange's Four Square Theorem.
		The development of Thue's Theorem is not normally considered part of a course in introductory number theory, but we have presented a short proof of it, and are able to use it effectively throughout.
		Complete solutions of Diophantine equation x^2-Dy^2=n for D>0 are given in via a non-standard approach using what we call semi-simple continued fractions.
	www.math.ucalgary.ca /~ramollin/fnt2.html (2355 words)

	Allouche, Peyrière, Wen, Wen, ...: Hankel determinants of the Thue-Morse sequence
		be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations:
		Together with three other sequences, it obeys a set of sixteen recurrence equations.
		THUE, Über die gegenseitige Lage gleicher Teile gewisse Zeichenreihen, Norske vid.
	www.numdam.org /numdam-bin/item?id=AIF_1998__48_1_1_0 (273 words)

	fwf5
		An essential part of this project is devoted to the effective solution of Dophantine equations from a theoretical as well as from an algorithmic and computational point of view.
		It will be dealt with specific Diophantine equations related to combinatorial questions, to classical orthogonal polynomials, Thue equations and index form equations.
		The results on polynomials which are necessary for the investigation of Diophantine equations are also a useful tool for exploring the distribution properties of polynomial and linear recurring sequences of algebraic integers modula a prime ideal.
	www.fsp83.jku.at /german/Projects.html (487 words)

	Solving elliptic Diophantine equations avoiding Thue equations and elliptic logarithms, Benjamin M. M. de Weger
		Solving elliptic Diophantine equations avoiding Thue equations and elliptic logarithms, Benjamin M. de Weger
		We determine the solutions in integers of the equation $ y^2 = ( x + p) ( x^2 + p^2) $ for $ p = 167$, $223$, $337$, $1201$.
		Bilu, and is shown to be in some cases more efficient than other general purpose methods for solving such equations, namely the elliptic logarithms method and the use of Thue equations.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1047674206 (93 words)

	Solving equations (Site not responding. Last check: 2007-11-06)
		The optional argument Exact can be used to indicate whether an exact solution is required (with Exact := true) or a solution up to a unit suffices.
		Given a monic irreducible polynomial of degree at least 3 over the integers, this function returns the `Thue object' corresponding to f; such object is used in the functions for solving Thue equations, and print as the homogeneous version of f.
		Given a Thue object t and an integer a this function return a sequence consisting of sequences of two integers [x, y] together providing all solutions to the (homogeneous form of) the Thue equation associated with t.
	www.math.ufl.edu /help/magma/text390.html (312 words)

	EconPapers: Integral and S-integral solutions of a Weierstrass equation
		Abstract: The rational solutions with as denominators powers of 2 of the elliptic diophantine equation y^2 = x^3 - 228 x + 848 are determined using three different methods.
		In the second method we deduce Thue and Thue-Mahler equations, and solve them using linear forms in real and p-adic logarithms, derived from three-term (S-) unit equations.
		Finally, a new idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term (S-) unit equations with special properties, that are again solved by linear forms in real and p-adic logarithms.
	netec.wustl.edu /WoPEc/data/Papers/dgreureir1997119.html (261 words)

	Publications1 (Site not responding. Last check: 2007-11-06)
		Pethõ und M. Pohst, On the resolution of index form equations in biquadratic number fields I. Number Theory, {\bf 38} (1991), 18-34.
		Peth\H o} and {\sc P. Voutier} {\em On the arithmetic of simplest sextic fields and related Thue equations,} In: "Number Theory", Eds.: K. Gy\H{o}ry, A.Peth\H{o} and V.T. S\'os, Walter de Gruyter GmbH \and Co., Berlin-New York, 1998, pp.
		Pethő, On the solutions of parametrized families of diophantine equations, Debrecen, October 25, 2001.
	www.inf.unideb.hu /~pethoe/Publications.html (2536 words)

	Mathematics of Computation
		An efficient algorithm is given for the resolution of relative Thue equations.
		Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields.
		I.Gaál and M.Pohst, On the resolution of index form equations in sextic fields with an imaginary quadratic subfield, J. Symbolic Comp.
	www.ams.org /mcom/2002-71-237/S0025-5718-01-01329-1/home.html (466 words)

	11: Number theory
		This has a direct bearing on other fields such as Diophantine equations, for example, since the unsolvability of a Diophantine equation can be deduced from the observation that it would require rational numbers which approximate a real number "too well".
		The sets of solutions in rational numbers to algebraic equations may be viewed as algebraic varieties, and thus studied with tools of 14: Algebraic Geometry.
		This is particularly true with single equations in two variables (which lead to curves); such equations when of degree 3 (or 4) lead to elliptic curves.
	www.math.niu.edu /~rusin/known-math/index/11-XX.html (2537 words)

	Diophantine Equations Number Theory Math Science English LoCuaL
		John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N. Linear Diophantine Equations
		MAGMA code to solve Diophantine equations of the form F(x)=G(y), for which Runge's condition is satisfied.
		Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
	locual.com /D/Science/Math/Number_Theory/Diophantine_Equations (428 words)

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