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Math Help - Congruence question

  1. #1
    Junior Member maths900's Avatar
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    Congruence question

    Congruences:

    Show by congruences that
    q^2 + 3r^2 + 6s^3 9p^5 = 2
    has no solutions in Z.
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  2. #2
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    Isomorphism's Avatar
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    Did you try reading the equation modulo three?
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  3. #3
    Junior Member maths900's Avatar
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    Quote Originally Posted by Isomorphism View Post
    Did you try reading the equation modulo three?
    How would i do that?
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  4. #4
    Senior Member vincisonfire's Avatar
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    In  \mathbb Z_3 the equation becomes q^2 =2 because 3,6 and 9 are in the equivalence class of 0.
    What are the squares in  \mathbb Z_3 ?
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  5. #5
    Junior Member maths900's Avatar
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    Quote Originally Posted by vincisonfire View Post
    In  \mathbb Z_3 the equation becomes q^2 =2 because 3,6 and 9 are in the equivalence class of 0.
    What are the squares in  \mathbb Z_3 ?
    Would it be plus or minus root 2?
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  6. #6
    Senior Member vincisonfire's Avatar
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     0^2=0
     1^2=1
     2^2=4=1
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  7. #7
    Junior Member maths900's Avatar
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    Quote Originally Posted by vincisonfire View Post
     0^2=0
     1^2=1
     2^2=4=1
    and these are not solutions in the set Z? sorry im new to number theory
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  8. #8
    Senior Member vincisonfire's Avatar
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    No problem. We all have to start somewhere. It will seem simple to you in few weeks from now.
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  9. #9
    Junior Member maths900's Avatar
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    Quote Originally Posted by vincisonfire View Post
    No problem. We all have to start somewhere. It will seem simple to you in few weeks from now.
    I hope so.

    Would i not be able to show it has no solutions in Z without using congruences??
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  10. #10
    Junior Member maths900's Avatar
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    Quote Originally Posted by vincisonfire View Post
    In  \mathbb Z_3 the equation becomes q^2 =2 because 3,6 and 9 are in the equivalence class of 0.
    What are the squares in  \mathbb Z_3 ?
    Here when you say q^2=2, do you mean q^2 is congruent to 2(mod3)?
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  11. #11
    Senior Member vincisonfire's Avatar
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    Yes, because there is no elements in mod3 such that its square is 2.
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