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Problem 422


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。