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Math Help - number theory

  1. #1
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    number theory

    evaluate legender symbol

    ( 2 / 2^43112609 - 1)

    ie ( 2/largest prime)
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    There is no largest prime.

    Use the fact that (2/p)=1 for p\equiv\pm 1 \mod 8 and (2/p)=-1 for p\equiv\pm 3 \mod 8.
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  3. #3
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    Out of curiosity, how would you evaluate linear congruences like
    2^{43112609} - 1 \equiv a \mbox{ (mod 8)} if you have BIG numbers, without a calculator or computer?
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  4. #4
    Super Member Bacterius's Avatar
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    If a \equiv b (mod c), then a^k \equiv b^k (mod c), I guess ?
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by ilikecandy View Post
    Out of curiosity, how would you evaluate linear congruences like
    2^{43112609} - 1 \equiv a \mbox{ (mod 8)} if you have BIG numbers, without a calculator or computer?
    Think about it : 2^n \equiv 0 \mod 8 for n \geq 3. So 2^{43112609} - 1 \equiv -1 \mod 8.
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