Difference between revisions of "Pell's equation (simple solutions)"
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==Equation of the form <math>x^2 – 2y^2 = 1</math>== | ==Equation of the form <math>x^2 – 2y^2 = 1</math>== | ||
+ | <math>\boldsymbol{a.}</math> Let integers <math>(x_i, y_i)</math> are the solution, <math>\hspace{10mm} x_i^2 - 2 y_i^2 = 1,</math> | ||
+ | <cmath>\begin{equation} \left\{ \begin{aligned} | ||
+ | x_{i+1} &= 3 x_i + 4 y_i ,\\ | ||
+ | y_{i+1} &= 2 x_i + 3 y_i . | ||
+ | \end{aligned} \right.\end{equation}</cmath> | ||
+ | then <math>x_{i+1}^2 - 2 y_{i+1}^2 = (3 x_i + 4 y_i)^2 - 2 (2 x_i + 3 y_i)^2 = x_i^2 - 2 y_i^2 = 1,</math> | ||
+ | |||
+ | therefore integers <math>(x_{i+1}, y_{i+1})</math> are the solution of the given equation. | ||
+ | <cmath>\{(x_i, y_i) \} = \{(1,0), (3,2), (17,12), (99,70),...\}.</cmath> |
Revision as of 02:46, 17 April 2023
Pell's equation is any Diophantine equation of the form where
is a given positive nonsquare integer, and integer solutions are sought for
and
Denote the sequence of solutions
It is clear that
During the solution we need:
a) to construct a recurrent sequence or two sequences
b) to prove that the equation has no other integer solutions.
Equation of the form ![$x^2 – 2y^2 = 1$](//latex.artofproblemsolving.com/2/7/9/27916919e274846741e72cf16f2b9b58113feea8.png)
Let integers
are the solution,
then
therefore integers are the solution of the given equation.