Show them the rational parametrization of the unit circle. Then generalize the method to other conics with an obvious rational point like x^2 + y^2 = 5 and x^2 - 2y^2 = 1 and discuss why it doesn't work on x^2 + y^2 = 3 (i.e., how would you show this conic has no rational points at all). Finally, discuss the parametrization of the integral points on x^2 - 2y^2 = 1, where the story is quite different from the case of rational points.
KConrad
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