Dictionary

Pell's equation en · NOUN

Etymology

Named by Leonhard Euler after the 17th-century mathematician John Pell, whom Euler mistakenly believed to be the first to find a general solution.

Meanings

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

Forms

Deriveds

generalised Pell's equation

Relateds

Pell number

Synonyms

Pell-Fermat equation

Translations (3)

de Pellsche Gleichung (Diophantine equation) · it equazione di Pell (Diophantine equation) · pl równanie Pella (Diophantine equation)