Let $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ be an ellipse. How can the number of integral points lying on such an ellipse be calculated ($A,B,C,D,E,F$ are, of course, integers) ?
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you could try all $(x,y)$ pairs... – tp1 Jun 22 '13 at 22:34
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4more to the point, Lagrange multipliers gives you bounds on $x,y,$ placing your ellipse in a rectangular box. You can try all $(x,y)$ integer pairs in the box. Or, you can vary $y$ over the indicated integers, use the quadratic formula for tight bounds on $x$ for each such $y.$ – Will Jagy Jun 22 '13 at 22:58
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The trial-and-error approaches mentioned above will work only if you're sure that you have an ellipse. The equation you gave can represent other types of conic section curves, too. In many cases, a conic section curve is unbounded, so you can't put a box around it. – bubba Jun 23 '13 at 00:21
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Do you only need to do this for a single ellipse or are you interested in the general problem? The general problem is essentially equivalent to counting the number of representations of an integer by a quadratic form and this is somewhat complicated in general (for example, the case $x^2 + y^2 = n$ is related to factorization in the Gaussian integers). In the non-ellipse case you get Pell's equations as special cases. – Qiaochu Yuan Jun 23 '13 at 01:42