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Math Help - Chinese Remainder Theorem

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    Chinese Remainder Theorem

    I have a system of congruences modulo different powers of three which satisfies the conditions of the Chinese Remainder Theorem. I have n+1 variables (all positive integers) and I have n+1 equations. For each k=1,2,...,n+1:

    \displaymode 2^{a_0}\equiv 1+\sum_{i=1}^{k-1}\left(3^i\left(\frac{3^{k}+1}{2}\right)^{\sum_{j  =1}^i {a_j}\right) (\text{mod }3^k)

    So, for example, if n=2, I have:

    \displaymode 2^{a_0}\equiv 1 (\text{mod }3)
    \displaymode 2^{a_0}\equiv 1+3\cdot 5^{a_1} (\text{mod }9)
    \displaymode 2^{a_0}\equiv 1+3\cdot 14^{a_1}+9\cdot 14^{a_1+a_2} (\text{mod }27)

    If I use the Chinese Remainder Theorem, I wind up with a system of nonlinear equations in \mathbb{Z} / 3^{n+1}\mathbb{Z}. Is this possible to solve for a_0 (\text{mod }2\cdot 3^n)?
    Last edited by SlipEternal; September 6th 2012 at 08:32 AM.
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