Pell's equation is the diophantine equation x squared minus n times y squared equals 1, where n is a positive integer that is not a perfect square, and it asks for integer solutions x and y. The Indian mathematician Brahmagupta found solutions to particular cases around 628 CE and developed a composition identity for generating new solutions from old ones, and the twelfth-century mathematician Bhaskara II gave the first general method, the chakravala technique, solving the equation for every value of n. Pierre de Fermat revived the problem in Europe in the 1650s, William Brouncker found solutions, Leonhard Euler mistakenly attached John Pell's name to the equation, and Joseph-Louis Lagrange developed the general theory using continued fractions between 1766 and 1769, proving that whenever n is not a perfect square, Pell's equation has infinitely many distinct integer solutions, all generated from a single fundamental solution.
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Source Pell's equation (Wikipedia)
Principal Result Of
Source Brahmagupta (MacTutor)
Proved By
Source Pell's equation (Wikipedia)
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1. Pell's equation (Wikipedia)
History
Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions.
Proved By: Joseph-Louis Lagrange, Lead paragraph
Lagrange proved that, as long as n is not a perfect square, Pell
In Branch: Number Theory, Lead section
passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and
View the SourceBrahmagupta (MacTutor)
Principal Result Of: Brahmagupta, Biography, indeterminate equationsQuote, Principal Result Of: Brahmagupta, Biography, indeterminate equations
Brahmagupta also solves quadratic indeterminate equations of the type
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