Difference between revisions of "Pell's equation (simple solutions)"
(→Equation of the form x^2 – 2y^2 = 1) |
(→Equation of the form x^2 – 2y^2 = 1) |
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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> | 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. | ||
+ | |||
+ | If <math>i > 0</math> then <math>x_{i+1} > y_{i+1} \ge 2(x_i + y_i) > 0.</math> | ||
+ | <cmath>\{(x_i, y_i) \} = \{(1,0), (3,2), (17,12), (99,70),...\}.</cmath> | ||
+ | <math>\boldsymbol{b.}</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. | therefore integers <math>(x_{i+1}, y_{i+1})</math> are the solution of the given equation. | ||
− | < | + | |
+ | If <math>x_i > 0, y_i > 0, x_{i+1} > 0, y_{i+1} > 0 </math> then <math>x_{i+1} > y_{i+1}, x_i > y_i .</math> |
Revision as of 03:46, 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.
If then
Let integers
are the solution,
then
therefore integers
are the solution of the given equation.
If then