Problem 403
Lattice points enclosed by parabola and line
For integers a and b, we define D(a, b) as the domain enclosed by the parabola y = x2 and the line y = a·x + b:
D(a, b) = { (x, y) | x2 ≤ y ≤ a·x + b }.
L(a, b) is defined as the number of lattice points contained in D(a, b).
For example, L(1, 2) = 8 and L(2, -1) = 1.
We also define S(N) as the sum of L(a, b) for all the pairs (a, b) such that the area of D(a, b) is a rational number and |a|,|b| ≤ N.
We can verify that S(5) = 344 and S(100) = 26709528.
Find S(1012). Give your answer mod 108.
被双曲线和直线包围的格点
对于正整数a和b，我们用D(a, b)表示被双曲线y = x2和直线y = a·x + b包围的区域：
D(a, b) = { (x, y) | x2 ≤ y ≤ a·x + b }.
定义L(a, b)为D(a, b)内的格点数目。
例如，L(1, 2) = 8，以及L(2, -1) = 1。
若整数对(a, b)满足D(a, b)的面积是有理数，且|a|,|b| ≤ N，我们定义所有相应的L(a, b)的和为S(N)。
我们可以验证S(5) = 344以及S(100) = 26709528。
求S(1012)。将你的答案模108取余。