Rotating the Hyperbola

Geometry Level 4

A hyperbola \(H\) with equation \(xy=n\) (where \(n\le 1000\)) is rotated \(45^{\circ}\) to obtain the hyperbola \(H'\). Let the positive difference between the number of lattice points on \(H\) and \(H'\) be \(D\). Given that both \(H\) and \(H'\) have at least one lattice point, find the maximum possible value of \(D\).

Details and Asumptions: A lattice point is a point that has integer \(x\)- and \(y\)-coordinates.

You may want to look at the list of Highly Composite Numbers.

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