Pell's equation

• Pell equation

Named by Leonhard Euler after the 17th-century mathematician John Pell, whom Euler mistakenly believed to be the first to find a general solution.[1]

Pell's equation (plural Pell's equations)

1. (number theory) The Diophantine equation ${\displaystyle x^{2}-my^{2}=1}$ for a given integer m, to be solved in integers x and y.
   • 1974, Allan M. Kirch, Elementary Number Theory: A Computer Approach, Intext Educational Publishers, page 212,

     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.

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

     Thus ${\displaystyle (P,Q)}$ satisfies Pell's equation and so by Lemma 1, ${\displaystyle P/Q}$ is a convergent to ${\displaystyle {\sqrt {d}}}$.

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

     We introduced Pell's equation

     ${\displaystyle \qquad \qquad \qquad \qquad x^{2}-ny^{2}=1}$

     in Chapter 4 as an example of a Diophantine equation. The solution ${\displaystyle x=577,\ y=408}$ of the Pell equation ${\displaystyle x^{2}-2y^{2}=1}$ was used in India in the fourth century to produce the fraction ${\displaystyle {\tfrac {577}{408}}}$ as an excellent rational approximation for ${\displaystyle {\sqrt {2}}}$.

     It is easy to see why solutions to Pell's equation can be used to approximate solutions to ${\displaystyle {\sqrt {n}}}$—this was known to Archimedes, who used this method to approximate square roots.

• Pell-Fermat equation

• generalised Pell's equation

• Pell number

• Pell's equation on Wikipedia.Wikipedia
• Pell number on Wikipedia.Wikipedia