Section 1: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Parametrise the conic $C:(x^{2}+y^{2}=5)$ by considering a variable line through $(2,1)$ and hence find all rational solutions of $x^{2}+y^{2}=5$ .

The parametrization is easily obtained as $\begin{eqnarray*} (x,y) & = & (2\tfrac{\lambda^{2}-\lambda-1}{\lambda^{2}+1},\tfrac{-\lambda^{2}-4\lambda+1}{\lambda^{2}+1}). \end{eqnarray*}$ Therefore, the rational solutions are given by $x=\tfrac{2l^{2}-2lm-2m^{2}}{l^{2}+m^{2}},y=\tfrac{-l^{2}-4lm+m^{2}}{l^{2}+m^{2}}$ for all $l,m\in\mathbb{Z}$ except $l=m=0$ , where one of them can be assumed to be positive, and they can be assumed to be relatively prime.

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Section 1: Problem 1 Solution

Section 1: Problem 1 Solution