Problem Archive

Let $H$ be the hyperbola defined by the equation $12x^2 + 7xy - 12y^2 = 625$.

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 $H$, $\{P_i: i \geq 1\}$, as:

• $P_1 = (13, 61/4)$.
• $P_2 = (-43/6, -4)$.
• For $i \gt 2$, $P_i$ is the unique point in $H$ that is different from $P_{i-1}$ and such that line $P_iP_{i-1}$ is parallel to line $P_{i-2}X$. It can be shown that $P_i$ is well-defined, and that its coordinates are always rational.

You are given that $P_3 = (-19/2, -229/24)$, $P_4 = (1267/144, -37/12)$ and $P_7 = (17194218091/143327232, 274748766781/1719926784)$.

Find $P_n$ for $n = 11^{14}$ in the following format:
If $P_n = (a/b, c/d)$ where the fractions are in lowest terms and the denominators are positive, then the answer is $(a + b + c + d) \bmod 1\,000\,000\,007$.

For $n = 7$, the answer would have been: $806236837$.