Download UPDATED Command And Conquer Red Alert 3 For Mac
Download Command And Conquer Red Alert 3 For Mac
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If you are interested in the Windows version of Red Alert 3, for example, I would recommend downloading .Q:
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
$$(u, v) \begin{pmatrix}19 & 82\\9 & 1\end{pmatrix}^{ -1} = (T_p(u, v), T_p(19 u, 2 v))$$
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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"An obvious ship has been seen in orbit in the area. A tall woman has been seen at the spaceport who fits the profile of a possible State Department operative but there's been no talk of an arrest. Darruthers and his party, the "White Rose", (headline of their paper) have printed their response to the State and the Philosophers' Collective: "We Are All Germinating Seeds." Four of them have been detained and moved to the mines. There have been rumours, unconfirmed by either side, that their trial will be meted out by a new Revolutionary Tribunal. "
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