1) $D$ is a positive integer, find all rational solutions of Pell's equation
$$x^2-Dy^2=1$$
2) What about $D\in\Bbb Q$ ?
Number Theory – Rational Solutions of Pell’s Equation
diophantine equationsnumber theorypell-type-equationsrational numbers
diophantine equationsnumber theorypell-type-equationsrational numbers
1) $D$ is a positive integer, find all rational solutions of Pell's equation
$$x^2-Dy^2=1$$
2) What about $D\in\Bbb Q$ ?
Best Answer
$$(\frac{x}y)^2-(\frac{1}y)^2=D,\\(\frac{x}y+\frac{1}y)(\frac{x}y-\frac{1}y)=D,\\\frac{x}y+\frac{1}y=t,\frac{x}y-\frac{1}y=\frac{D}t,\\\frac{x}y=\frac{1}2(t+\frac{D}{t}),\frac{1}y=\frac{1}2(t-\frac{D}{t}),\\x=\dfrac{t^2+D}{t^2-D},y=\frac{2t}{t^2-D},(t\in \mathbb Q,t^2\neq D).$$