solve x^2-29y^2=1 where x and y are integers and y is not zero
The equation ( x^2 - 29y^2 = 1 ) is a Pell's equation of the form ( x^2 - Dy^2 = 1 ), where ( D = 29 ). To solve this equation, we can find the fundamental solution and then generate further solutions using the properties of Pell's equations.
Find the fundamental solution: The fundamental solution of the Pell's equation ( x^2 - 29y^2 = 1 ) can be found using continued fractions or by trial and error. The fundamental solution for ( D = 29 ) is ( (x_1, y_1) = (169, 31) ).
Generate further solutions: The solutions to the Pell's equation can be generated using the recurrence relations based on the fundamental solution: [ x_{n+1} = x_1 x_n + 29 y_1 y_n ] [ y_{n+1} = x_1 y_n + y_1 x_n ] Starting with the fundamental solution ( (x_1, y_1) = (169, 31) ), we can generate more solutions.
Calculate additional solutions:
For ( n = 1 ): [ x_2 = 169 \cdot 169 + 29 \cdot 31 \cdot 31 = 28561 + 29 \cdot 961 = 28561 + 27869 = 56430 ] [ y_2 = 169 \cdot 31 + 31 \cdot 169 = 5239 + 5239 = 10478 ] So, the second solution is ( (56430, 10478) ).
For ( n = 2 ): [ x_3 = 169 \cdot 56430 + 29 \cdot 31 \cdot 10478 ] [ y_3 = 169 \cdot 10478 + 31 \cdot 56430 ] This will yield larger integers.
List the solutions: The first few integer solutions ( (x, y) ) to the equation ( x^2 - 29y^2 = 1 ) are:
Conclusion: The integer solutions to the equation ( x^2 - 29y^2 = 1 ) with ( y \neq 0 ) include ( (169, 31) ), ( (56430, 10478) ), and more can be generated using the recurrence relations.