Difference between revisions of "Pell's equation (simple solutions)"
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Pell's equation is any Diophantine equation of the form <math>x^2 – Dy^2 = 1,</math> where <math>D</math> is a given positive nonsquare integer, and integer solutions are sought for <math>x</math> and <math>y.</math> | Pell's equation is any Diophantine equation of the form <math>x^2 – Dy^2 = 1,</math> where <math>D</math> is a given positive nonsquare integer, and integer solutions are sought for <math>x</math> and <math>y.</math> | ||
| − | Denote the sequence of solutions <math>{x_i, y_i}. </math> | + | Denote the sequence of solutions <math>\{x_i, y_i \}. </math> |
| − | It is clear that <math>{x_0, y_0} = {1,0}.</math> | + | It is clear that <math>\{x_0, y_0 \} = \{1,0 \}.</math> |
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During the solution we need: | During the solution we need: | ||
| − | a) to construct a recurrent sequence <math>{x_{i+1}, y_{i+1}} = f({x_i, y_i})</math> or two sequences <math>{x_{i+1}} = f({x_i}), {y_{ i+1}} = g({y_i});</math> | + | a) to construct a recurrent sequence <math>\{x_{i+1}, y_{i+1} \} = f({x_i, y_i})</math> or two sequences <math>\{x_{i+1} \} = f({x_i}), \{y_{ i+1} \} = g({y_i});</math> |
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b) to prove that the equation has no other integer solutions. | b) to prove that the equation has no other integer solutions. | ||
==Equation of the form <math>x^2 – 2y^2 = 1</math>== | ==Equation of the form <math>x^2 – 2y^2 = 1</math>== | ||
Revision as of 15:48, 16 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.
