Problem 422
Sequence of points on a hyperbola
Let H be the hyperbola defined by the equation 12x2 + 7xy - 12y2 = 625.
Next, define X as the point (7, 1). It can be seen that X is in H.
Now we define a sequence of points in H, {Pi : i ≥ 1}, as:
- P1 = (13, 61/4).
- P2 = (-43/6, -4).
- For i > 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi is well-defined, and that its coordinates are always rational.
You are given that P3 = (-19/2, -229/24), P4 = (1267/144, -37/12) and P7 = (17194218091/143327232, 274748766781/1719926784).
Find Pn for n = 1114 in the following format:
If Pn = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.
For n = 7, the answer would have been: 806236837.
双曲线上的点序列
双曲线H由方程12x2 + 7xy - 12y2 = 625给出。
取点X的坐标为(7, 1)，可以看出点X在H上。
接下来我们定义H上的一个点序列{Pi : i ≥ 1}如下：
- P1 = (13, 61/4)。
- P2 = (-43/6, -4)。
- 对于i > 2，Pi是在H上异于Pi-1的一点，且满足直线 PiPi-1与直线 Pi-2X平行。可以验证Pi是存在的，且它的坐标总是有理数。
已知P3 = (-19/2, -229/24)，P4 = (1267/144, -37/12)以及P7 = (17194218091/143327232, 274748766781/1719926784)。
当n = 1114时，求Pn，并以如下形式给出：
如果Pn = (a/b, c/d)，其中的分数均为最简分数且分母为正整数，那么给出的答案应当是(a + b + c + d) mod 1 000 000 007。
例如，当n = 7时，答案应当是：806236837。