Problem 404

Problem 404

Crisscross Ellipses

Ea is an ellipse with an equation of the form x2 + 4y2 = 4a2.
Ea‘ is the rotated image of Ea 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(103) = 7, C(104) = 106 and C(106) = 11845.

Find C(1017).


Ea是一个椭圆,其方程为x2 + 4y2 = 4a2
Ea‘是将Ea绕原点O(0, 0)逆时针旋转角θ得到的图形,其中0° < θ < 90°。

如果有序三元组(a, b, c)中a、b和c均为正整数,我们称之为规范椭圆三元组。
例如,(209, 247, 286)就是一个规范椭圆三元组。

记C(N)为所有满足a ≤ N的规范椭圆三元组(a, b, c)的数目。
可以验证C(103) = 7,C(104) = 106,以及C(106) = 11845。