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Topic: Diophantine equation

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Diophantine equation

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	Diophantine equation - Wikipedia, the free encyclopedia
		A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.
		The central idea of Diophantine geometry is that of a rational point, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed.
		The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.
	en.wikipedia.org /wiki/Diophantine_equation (516 words)

	11D: Diophantine equations (Site not responding. Last check: 2007-11-06)
		Diophantine equations whose solution set is one-dimensional are discussed with algebraic curves.
		Equations whose solutions are curves of genus 1 are discussed in the subsection on elliptic curves.
		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 (775 words)

	PlanetMath: Diophantine equation
		A Diophantine equation is an equation for which the solutions are required to be integers.
		Generally, solving a Diophantine equation is not as straightforward as solving a similar equation in the real numbers.
		This is version 3 of Diophantine equation, born on 2001-11-16, modified 2002-02-27.
	planetmath.org /encyclopedia/DiophantineEquation.html (270 words)

	Equation - Wikipedia, the free encyclopedia
		An equation is a mathematical statement, in symbols, that two things are the same.
		Equations are often used to state the equality of two expressions containing one or more variables.
		However, if the equation were based on the natural numbers for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed.
	en.wikipedia.org /wiki/Equation (544 words)

	diophanfin.html (Site not responding. Last check: 2007-11-06)
		Diophantine equations are equations of polynomial expressions for which rational or integer solutions are sought.
		If a given equation has rational solutions, a corresponding equation with integer solutions can be found by multiplying the first equation by an integer constant, namely, the least common multiple of the denominators of the numbers obtained by raising the solutions to the appropriate power.
		For Diophantine equations of the type ax+by=c, there exists an infinite number of solutions if (a,b)c, that is if the greatest common divisor of a and b divides c.
	www.ms.uky.edu /~carl/ma330/projects/diophanfin1.html (2434 words)

	Diophantine equation -- Encyclopædia Britannica (Site not responding. Last check: 2007-11-06)
		equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers.
		Equations that involve nonalgebraic operations, such as the evaluation of logarithms or trigonometric functions, are said to be transcendental.
		Collection of equations involving only addition, multiplication, or taking powers in which all the constants are natural numbers or their negatives and the only solutions of interest are natural numbers or their negatives.
	www.britannica.com /eb/article-9030556 (785 words)

	Equation (Site not responding. Last check: 2007-11-06)
		Although the basic theory of structural equation modeling is an extension of single- equation regression concepts, the matrix language that is typically...
		In equations, the values of the variables for which the equation is true are called solutions.
		Thus to solve the equation, one must express those values of the variables that are solutions in terms of whatever constants may be included.
	www.hallencyclopedia.com /Equation (446 words)

	[No title]
		Other types of Diophantine problems with (partial) solutions The basic setting for Diophantine problems is a set of equations which are to be solved in whole numbers (or in some cases rational numbers or numbers of a certain form).
		Whenever some of the equations defining a variety are linear, we may usually take these to define some of the variables in terms of the other; that is, we perform a projection from a space of higher dimension to a space of lower dimension using a projection which is almost everywhere a one-to-one correspondence.
		Indeed, if the quadratic equation is written Q + L - c = 0, where Q is a homogeneous quadratic form, L is linear in the X_i, and c is a constant, then the construction of a linear transformation to render Q diagonal is the usual Gram-Schmidt process, without the extraction of square roots.
	www.math.niu.edu /~rusin/known-math/98/diophantine (4373 words)

	Section 1.1 from "Hilbert's tenth problem" by Yuri MATIYASEVICH
		When we speak of ''an arbitrary Diophantine equation,'' we shall have in mind an equation of the form (1.1.1) since an equation of the form (1.1.2) can easily be transformed into an equation of the form (1.1.1) by transposing all the terms to the left-hand side.
		Diophantine equations typically have several unknowns, and we must distinguish the degree of (1.1.1) with respect to a given unknown x
		In specifying a Diophantine equation it is necessary to provide not only a representation of the form (1.1.1) (or equivalently (1.1.2)), but also to indicate the range of the unknowns.
	logic.pdmi.ras.ru /~yumat/H10Pbook/par_1_1.htm (718 words)

	Number Theory: Diophantine Equations (Site not responding. Last check: 2007-11-06)
		A Diophantine equation is an algebraic equation in one or more unknowns with integer coefficients, for which integer solutions are sought.
		The simplest case is the linear Diophantine equation in two unknowns:
		Equation (1) has a solution if and only if c is a multiple of (a, b).
	www.uz.ac.zw /science/maths/zimaths/62/dioph.htm (565 words)

	Diophantine Equations (Site not responding. Last check: 2007-11-06)
		In general, the above "algebraic equations in the domain of integer numbers" can be defined as P=0, where P is a polynomial with integer coefficients and one, two or more variables (the "unknowns").
		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).
	www25.brinkster.com /ranmath/diophan/diophan.htm (1883 words)

	Diophantine set - Wikipedia, the free encyclopedia
		(Such a polynomial equation over the integers is also called a Diophantine equation.) In other words, a Diophantine set is a set of the form
		Matiyasevich's theorem, published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable.
		A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever.
	en.wikipedia.org /wiki/Diophantine_set (158 words)

	Open Directory - Science: Math: Number Theory: Diophantine Equations (Site not responding. Last check: 2007-11-06)
		Diophantine m-tuples - Sets with the property that the product of any two distinct elements 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.
		Solving General Pell Equations - John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N. Thue Equations - Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
	dmoz.org /Science/Math/Number_Theory/Diophantine_Equations (454 words)

	Bibi Drum
		Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
		A Diophantine equation may extend from a first-degree polynomial to a polynomial of an infinite degree.
		If you can come up with an equation that is hard to solve in a finite number of steps, you can use that equation to code in different commands and information that you do not want the general public to know.
	www.mathsci.appstate.edu /~sjg/wmm/student/robinson/robinsonp.htm (1636 words)

	Historical Notes: Diophantine equations
		There is a fairly complete theory of homogeneous quadratic Diophantine equations with three variables, and on the basis of results from the early and mid-1900s a finite procedure should in principle be able to handle quadratic Diophantine equations with any number of variables.
		In 1909 Axel Thue showed that any equation of the form p[x,y] == a, where p[x, y] is a homogeneous irreducible polynomial of degree at least 3 (such as x^3+x y^2 + y^3) can have only a finite number of integer solutions.
		If one wants to enumerate all possible Diophantine equations there are many ways to do this, assigning different weights to numbers of variables, and sizes of coefficients and of exponents.
	www.wolframscience.com /reference/notes/1164b (1183 words)

	Historical Notes: Hilbert's tenth problem
		The notion that there might be universal Diophantine equations for which Hilbert’s Tenth Problem would be fundamentally unsolvable emerged in work by Martin Davis in 1953.
		It had been known since the 1930s that any Diophantine equation can be reduced to one with degree 4 - and in 1980 James Jones showed that a universal Diophantine equation with degree 4 could be constructed with 58 variables.
		It is even conceivable that a Diophantine equation with 2 variables could be universal: with one variable essentially being used to represent the program and input, and the other the execution history of the program - with no finite solution existing if the program does not halt.
	www.wolframscience.com /reference/notes/1161a (362 words)

	Diophantine equation
		An equation that has integer coefficients and for which integer solutions are required.
		One of the challenges (the tenth one) that David Hilbert threw down to twentieth-century mathematicians in his famous list was to find a general method for solving equations of this type.
		In 1970, however, the Russian mathematician Yuri Matiyasevich showed that there is no general algorithm for determining whether a particular Diophantine equation is soluble: the problem is undecidable.
	www.daviddarling.info /encyclopedia/D/Diophantine_equation.html (203 words)

	Diophantus (Site not responding. Last check: 2007-11-06)
		Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.
		Equations which would lead to solutions which are negative or
		We began this article with the remark that Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics.
	www-history.mcs.st-andrews.ac.uk /Mathematicians/Diophantus.html (2283 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 (4250 words)

	3. The Problem of Simplest Diophantine Representation
		Obviously, because of Theorem 1, every finite set has a Diophantine representation; moreover, among Diophantine equations, there must be a minimal complexity such that there is an equation with this complexity that represents a given set, and that no simpler equation does.
		We may agree that in such a case any one of the simplest equations will do; or, alternatively, we may further define an ``alphabetical'' ordering of the basic symbols, and agree that given a set of equations of the same complexity, by ``the simplest'' one means the first equation in the alphabetical order.
		Now assume that S is any finite set such that the complexity of the Diophantine equation P that is in fact the simplest one that provides a Diophantine representation of S is greater than c.
	www.hf.uio.no /filosofi/njpl/vol2no2/diophantine/node3.html (827 words)

	Diophantine Equation and Eigenvalue Problem (Site not responding. Last check: 2007-11-06)
		There is significant resemblance between Diophantine equation and eigenvalue problem on partial differential equation.
		It is known in general that the method of successive approximation is effective to solve this kind of integral equation when the value of kernel or Fredholm's determinant is properly small.
		From the similar discussion to the case of one-dimensional equation, we know that h' converges in the mean if and only if the function f doesn't contain the eigenfunction corresponding to the eigenvalue = 0 and the series S(f) is bounded.
	www.users.bigpond.com /tiddler2/c27508/diophantus.htm (930 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		Date: 02/05/2003 at 11:00:20 From: Jane Subject: Linear Diophantine equation Given positive integers a and b such that ab^2, b^2a^3, a^3b^4, b^4a^5,....., prove that a = b.
		Date: 02/06/2003 at 09:56:16 From: Doctor Jacques Subject: Re: Linear Diophantine equation Hi Jane, If both a and b are equal to 1, we certainly have a=b.
		If we divide the last equation by 2n, we find: j - (j/2n) <= i As this relation must be true for all n, we may choose n > j.
	mathforum.org /library/drmath/view/62186.html (256 words)

	Math Forum - Ask Dr. Math
		Now there are two cases, if the right side of this equation is zero or not.
		This boils down to finding the solutions of either of two linear Diophantine equations in two unknowns: (2Ak)x + (Bk-k^2)y = -Dk + 2AE - BD, or (2Ak)x + (Bk+k^2)y = -Dk - 2AE + BD. Any solution to either equation will be a solution to the original one.
		A necessary and sufficient condition for the existence of a solution to one of these equations is that the GCD of the coefficients of x and y be a divisor of the right-hand side.
	mathforum.org /library/drmath/view/55988.html (655 words)

	DIOPHANTINE EQUATION (Site not responding. Last check: 2007-11-06)
		In number-theoretical literature Diophantine equations are sometime treated in a broader sense by taking into account only the domain of admissible values of the unknowns.
		In such a case no distinction is made, for example, between the genuine Diophantine equations and so-called exponential Diophantine equations.
		Usually they are allowed to be arbitrary integers, but when the domain of unknowns is broader than the set of integers, it is sometimes natural to allow the coefficients to belong to the set of admissible values of the unknowns.
	logic.pdmi.ras.ru /Hilbert10/glossary/de.html (273 words)

	The Prime Glossary: Diophantus (Site not responding. Last check: 2007-11-06)
		The problems he worked on were mostly linear systems of equations with a few quadratics.
		Now we call an equation to be solved in integers a diophantine equation.
		This equation is solvable if and only if the greatest common divisor of a and b divides c.
	primes.utm.edu /glossary/page.php?sort=Diophantus (176 words)

	BBC - h2g2 - Diophantine Equations
		With only one equation, absolutely any value of x can be used and a corresponding value of y can be calculated, which is not normally very useful.
		In this sort of equation, there is a restriction that all the numbers involved must be whole numbers without any fractional or decimal part.
		This is a simpler equation to solve and it has already been shown how to solve it: one solution is z=3, x=34.
	www.bbc.co.uk /dna/h2g2/A964956 (1437 words)

	Diophantine Equations Project
		Explain how Diophantine equations are related to finding lattice points on lines.
		Explain how the question “does the rth row of the multiplication table mod N contain a number k” leads to a Diophantine equation.
		Theorem: The Diophantine equation ax + by = c has a solution if, and only if gcd(a,b) divides c.
	www.math.neu.edu /~bridger/U170/Diophantine.htm (260 words)

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