If an ellipse is centered on the origin with verticies (±a,0) and (0,±b), it may pass through some points with integer coordinates. In fact, by symmetry this will be a multiple of 2.

For example with a=4 and b=2√3 the ellipse passes through the 6 points (±4,0) and (±2,±3).

Find the smallest ellipse (in terms of area = πab) with a>b that passes through 2n integer points where n=2,3,4,5,6.

Feel free to go further.

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