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Prove that for any $n\geq 0$, there is a hyperbola that passes through exactly $n$ lattice points (= points with integer coordinates) and find an example.

For example it is easy to see that the hyperbola $xy=1$ passes throught exactly two lattice points. It is also easy to see that the hyperbola $$ x^2-2y^2=1 $$ passes through infinitely many lattice points, because this is a Pell equation and its integer solutions are related to the units of the ring of algebraic integers. The intermediate case of a prescribed number of lattice points requires another idea.

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    I removed some good comments as obsolete because they referred to an earlier version of the question.2014-12-29
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    [Associated meta post.](http://meta.math.stackexchange.com/q/18960/242)2014-12-29

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