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Math Help - Chinese Remainder Theorem

  1. #1
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    Chinese Remainder Theorem

    Hi

    Hi,

    Could anyone help with this proof? I need to use the chinese remainder theorem, but have no idea where to start? Does anyone have any ideas?

    To check if f(x, y) = 0 (mod m) has integer solutions where
    m > 1 and m in Z. It suffces to check it for prime powers, where m = p^k where k is bigger than or greater than 1
    and where p is a prime.

    Any ideas or hints would be great!

    Thanks!
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  2. #2
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    Quote Originally Posted by GraceMahon View Post
    Hi

    Hi,

    Could anyone help with this proof? I need to use the chinese remainder theorem, but have no idea where to start? Does anyone have any ideas?

    To check if f(x, y) = 0 (mod m) has integer solutions where
    m > 1 and m in Z. It suffces to check it for prime powers, where m = p^k where k is bigger than or greater than 1
    and where p is a prime.

    Any ideas or hints would be great!

    Thanks!
    Look at the statement of the theorem, and use the fact that p_i^{k_i} are pairwise coprime. If you don't see it, try looking at some smaller numbers (for example, take any multiple of 2^3 * 3^2 * 5 * 7 and then look at the congruence mod (2^3), mod (3^2), etc.). You should see a very definite pattern.
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