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.