Problem 404

Crisscross Ellipses

E_{a} is an ellipse with an equation of the form x^{2} + 4y^{2} = 4`a`^{2}.

E_{a}' is the rotated image of E_{a} by θ degrees counterclockwise around the origin O(0, 0) for 0° θ 90°.

`b` is the distance to the origin of the two intersection points closest to the origin and `c` is the distance of the two other intersection points.

We call an ordered triplet (`a`, `b`, `c`) a *canonical ellipsoidal triplet* if `a`, `b` and `c` are positive integers.

For example, (209, 247, 286) is a canonical ellipsoidal triplet.

Let C(`N`) be the number of distinct canonical ellipsoidal triplets (`a`, `b`, `c`) for `a` `N`.

It can be verified that C(10^{3}) = 7, C(10^{4}) = 106 and C(10^{6}) = 11845.

Find C(10^{17}).

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