How Low Can You Go?

Consider an ellipse with the following equation:

\[ax^{2}+by^{2}=100\]

How many lattice points within the region \(-1000 \leq x ,y\leq 1000\) cannot lie on any ellipse with real numbers \(a,b\) satisfying \(0<a \leq b \leq 10\)?

Details and Assumptions:

  • A lattice point is defined as a point with integer coordinates.

  • \(a\) and \(b\) are not necessarily integers, just real numbers within the constraint.

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