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A lattice point in a plane is one both of whose coordinates are integers. Let O be $$(\sqrt{2}, 1)$$ and P any given lattice point. Then the number of lattice points Q, distinct from P, such that $$OP = OQ$$ is
Let $$P = (p, q)$$ and $$Q = (r, s)$$. $$OP = OQ$$ gives $$r^2 + s^2 - 2s - 2\sqrt{2}\,r = p^2 + q^2 - 2q - 2\sqrt{2}\,p$$, and since $$\sqrt{2}$$ is irrational, $$r = p$$ and $$(s - q)(s + q - 2) = 0$$. So $$Q = (p, 2 - q)$$, which differs from P unless $$q = 1$$. The number is 1 in general and 0 when $$q = 1$$.
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