Pell's equation - Weblio 英和・和英辞典 (original) (raw)

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意味・対訳ペル方程式（ペルほうていしき、英: Pell's equation）とは、n を平方数ではない自然数として、未知整数 x、y についての形のディオファントス方程式である。

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Pell's equation

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Pell's equation

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Pell's equation

固有名詞

Pell 's equation

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