In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only...
33 KB (4,801 words) - 17:22, 11 August 2024
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a...
12 KB (1,475 words) - 01:09, 29 June 2024
Hilbert's tenth problem (category Diophantine equations)
given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution...
24 KB (3,132 words) - 16:56, 17 January 2024
five sextic equation for degree six septic equation for degree seven octic equation for degree eight A Diophantine equation is an equation where the unknowns...
32 KB (4,237 words) - 17:01, 26 August 2024
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x 2 − n y 2 = 1 , {\displaystyle x^{2}-ny^{2}=1,} where...
48 KB (6,613 words) - 01:05, 25 August 2024
Fermat's Last Theorem (section Diophantine equations)
Claude Bachet. Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2...
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composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according...
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equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation...
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be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a...
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In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in...
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example, consider the equation x 2 = 2. {\displaystyle x^{2}=2.} This equation can be viewed as a Diophantine equation, that is, an equation for which only integer...
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In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became...
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Brahmagupta (section Diophantine analysis)
the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as...
44 KB (5,847 words) - 07:01, 5 September 2024
algebraic equation over the rationals. A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is...
14 KB (2,162 words) - 02:58, 9 October 2023
Number theory (section Diophantine geometry)
Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood...
88 KB (11,254 words) - 14:06, 23 July 2024
Cannonball problem (category Diophantine equations)
to America. Édouard Lucas formulated the cannonball problem as a Diophantine equation ∑ n = 1 N n 2 = M 2 {\displaystyle \sum _{n=1}^{N}n^{2}=M^{2}} or...
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Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical...
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in Wiktionary, the free dictionary. Diophantine approximation Diophantine equation Diophantine quintuple Diophantine set This disambiguation page lists...
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Bézout's identity (redirect from Linear diophantine equation)
identity for homogeneous polynomials in three indeterminates Diophantine equation – Polynomial equation whose integer solutions are sought Euclid's lemma – A...
12 KB (1,639 words) - 13:35, 1 September 2024
The monkey and the coconuts (category Diophantine equations)
class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some...
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Diophantus and Diophantine Equations is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus...
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Pythagorean triple (category Diophantine equations)
the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. There...
81 KB (11,397 words) - 20:39, 1 September 2024
In mathematics, a Thue equation is a Diophantine equation of the form f ( x , y ) = r , {\displaystyle f(x,y)=r,} where f {\displaystyle f} is an irreducible...
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Diophantus (section Diophantine analysis)
Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations...
27 KB (3,318 words) - 09:21, 25 July 2024
applications in the study of Diophantine equations. The 2022 Fields Medal was awarded to James Maynard for his work on Diophantine approximation. Given a real...
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Brocard's problem (redirect from Brocard–Ramanujan Diophantine equation)
the Diophantine equation x! + A = y2", Nieuw Archief voor Wiskunde, 14 (3): 321–324, MR 1430045 Luca, Florian (2002), "The Diophantine equation P(x)...
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both sides by abc shows that the optic equation is equivalent to a Diophantine equation (a polynomial equation in multiple integer variables). All solutions...
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Proof of impossibility (section Integer solutions of Diophantine equations: Hilbert's tenth problem)
not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an...
29 KB (3,909 words) - 08:17, 2 August 2024
for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by = c where x and y are...
21 KB (2,502 words) - 01:19, 26 August 2024
Vieta jumping (category Diophantine equations)
solutions to an equation using Vieta's formulas. Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic...
15 KB (1,858 words) - 14:08, 17 February 2024