Pell's equation

English[edit]

Alternative forms[edit]

Pell equation

Etymology[edit]

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]

Noun[edit]

Pell 's equation (plural Pell's equations)

(number theory) The Diophantine equation $x^{2}-my^{2}=1$ $x^{2}-my^{2}=1$ 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 $(P,Q)$ satisfies Pell's equation and so by Lemma 1, $P/Q$ is a convergent to ${\sqrt {d}}$ .
- 2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409:
  We introduced Pell's equation
  $\qquad \qquad \qquad \qquad x^{2}-ny^{2}=1$
  in Chapter 4 as an example of a Diophantine equation. The solution $x=577,\ y=408$ of the Pell equation $x^{2}-2y^{2}=1$ was used in India in the fourth century to produce the fraction ${\tfrac {577}{408}}$ as an excellent rational approximation for ${\sqrt {2}}$ .
  It is easy to see why solutions to Pell's equation can be used to approximate solutions to ${\sqrt {n}}$ —this was known to Archimedes, who used this method to approximate square roots.