# Download UPDATED Command And Conquer Red Alert 3 For Mac

## Download UPDATED Command And Conquer Red Alert 3 For Mac

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solving system of Pell’s equation

I am trying to solve $19^x + 9^y = 82 \mod p$ where $p$ is a prime number and $x, y \in \mathbb{Z}$.
From the comments below my previous question (solving Pell’s equation with Bézout’s identity and Hacker’s method), I have found there is a quadratic relation between $x$ and $y$.
If $\bar{u} = (u, v)$ is the vector of $u$ and $v$, then I have found
$$(x, y) = (u, v) A^{ -1} = T_p(19^x, 9^y)$$
where $A = \begin{pmatrix}82 & 19\\9 & 1\end{pmatrix}$ is the matrix of $T_p$ of rank 2 and it has no repeated eigenvalues.
By the way, $T_p$ is the transformation that induces a linear recursion between $x$ and $y$, and the recursion is not polynomial. So I am looking for the general solution of $19^x + 9^y = 82 \mod p$ (for all primes $p$). Is it possible? Thanks in advance.
EDIT: $T_p$ is the matrix induced by
$$(u, v) \begin{pmatrix}19 & 82\\9 & 1\end{pmatrix}^{ -1} = (T_p(u, v), T_p(19 u, 2 v))$$

A:

Your matrix $T_p$ is not going to work. It is not invertible since it has repeated eigenvalues $\lambda=1$ and $\lambda=82$ and therefore eigenvalues which
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