Meaning of Pell's equation | Babel Free

Definitions

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

Equivalents

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.”

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

“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.”

CEFR level