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- Pell's_equation abstract "Pell's equation is any Diophantine equation of the formwhere n is a given nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.The name of Pell's equation arose from Leonhard Euler's mistakenly attributing its study to John Pell. Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell. This equation was first studied extensively in ancient India, starting with Brahmagupta, who developed the chakravala method to solve Pell's equation and other quadratic indeterminate equations in his Brahma Sphuta Siddhanta in 628, about a thousand years before Pell's time. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was subsequently translated into Latin in 1126. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Solutions to specific examples of the Pell equation, such as the Pell numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and to a similar date in India.For a more detailed discussion of much of the material here, see Lenstra (2002) and Barbeau (2003).".
- Pell's_equation thumbnail Pell's_equation.svg?width=300.
- Pell's_equation wikiPageExternalLink phd.pdf.
- Pell's_equation wikiPageExternalLink ABV2773.0001.001.
- Pell's_equation wikiPageExternalLink Pell.html.
- Pell's_equation wikiPageExternalLink fea-lenstra.pdf.
- Pell's_equation wikiPageID "24877".
- Pell's_equation wikiPageRevisionID "604356413".
- Pell's_equation hasPhotoCollection Pell's_equation.
- Pell's_equation subject Category:Continued_fractions.
- Pell's_equation subject Category:Diophantine_equations.
- Pell's_equation type Abstraction100002137.
- Pell's_equation type Communication100033020.
- Pell's_equation type ComplexNumber113729428.
- Pell's_equation type ContinuedFraction113736550.
- Pell's_equation type ContinuedFractions.
- Pell's_equation type DefiniteQuantity113576101.
- Pell's_equation type DiophantineEquations.
- Pell's_equation type Equation106669864.
- Pell's_equation type Fraction113732078.
- Pell's_equation type MathematicalStatement106732169.
- Pell's_equation type Measure100033615.
- Pell's_equation type Message106598915.
- Pell's_equation type Number113582013.
- Pell's_equation type RationalNumber113730469.
- Pell's_equation type RealNumber113729902.
- Pell's_equation type Statement106722453.
- Pell's_equation comment "Pell's equation is any Diophantine equation of the formwhere n is a given nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions.".
- Pell's_equation label "Ecuación de Pell".
- Pell's_equation label "Equazione di Pell".
- Pell's_equation label "Pell's equation".
- Pell's_equation label "Pellsche Gleichung".
- Pell's_equation label "Równanie Pella".
- Pell's_equation label "Vergelijking van Pell".
- Pell's_equation label "Équation de Pell-Fermat".
- Pell's_equation label "Уравнение Пелля".
- Pell's_equation label "ペル方程式".
- Pell's_equation label "佩尔方程".
- Pell's_equation sameAs Pellsche_Gleichung.
- Pell's_equation sameAs Ecuación_de_Pell.
- Pell's_equation sameAs Équation_de_Pell-Fermat.
- Pell's_equation sameAs Equazione_di_Pell.
- Pell's_equation sameAs ペル方程式.
- Pell's_equation sameAs 펠_방정식.
- Pell's_equation sameAs Vergelijking_van_Pell.
- Pell's_equation sameAs Równanie_Pella.
- Pell's_equation sameAs m.066zb.
- Pell's_equation sameAs Q853067.
- Pell's_equation sameAs Q853067.
- Pell's_equation sameAs Pell's_equation.
- Pell's_equation wasDerivedFrom Pell's_equation?oldid=604356413.
- Pell's_equation depiction Pell's_equation.svg.
- Pell's_equation isPrimaryTopicOf Pell's_equation.