Pell's equation

plural Pell's equations · alternative Pell equation

1. mathematicsThe Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y.

   Thus (P,Q) satisfies Pell's equation and so by Lemma 1, P#47;Q is a convergent to #92;sqrtd.

   1989, Mathematics Magazine, Volume 62, Mathematical Association of America, page 258

   2 more examples

   However, due to Euler's mistake in attributing the equation to English mathematician John Pell (1610-1585), Equation (27.14) is called Pell's equation. Results concerning Pell's equation will be stated without proof.

   1974, Allan M. Kirch, Elementary Number Theory: A Computer Approach, Intext Educational Publishers, page 212

   We introduced Pell's equation #92;qquad#92;qquad#92;qquad#92;qquadx²-ny²#61;1 in Chapter 4 as an example of a Diophantine equation. The solution x#61;577,#92;y#61;408 of the Pell equation x²-2y²#61;1 was used in India in the fourth century to produce the fraction #92;tfrac#123;577#125;#123;408#125; as an excellent rational approximation for #92;sqrt 2. It is easy to see why solutions to Pell's equation can be used to approximate solutions to #92;sqrtn—this was known to Archimedes, who used this method to approximate square roots.

   2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409

   Synonyms: Pell-Fermat equation