perplexus.info :: Geometry : Integer points on ellipses

Integer points on ellipses (Posted on 2011-03-10)

If an ellipse is centered on the origin with verticies (ąa,0) and (0,ąb), 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 (ą4,0) and (ą2,ą3).

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.

Submitted by Jer

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Solution: (Hide)

n=1, a=1, b→0, A=0, points (±1,0)
n=2, a=4, b=1, A=4pi, (±2,0), (±13,±1)
n=3, a=12, b=4, A=48pi, (0,±2), (±3,±1)
n=4, a=20, b=5, A=100pi, (±2,±2), (±4,±1)
n=5, a=162, b=81, A=13112pi, (0,±9), (±8,±7), (±12,±3)
n=6, a=156, b=52, A=8112pi, (±3,±7), (±9,±5), (±12,±2)

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