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Math Help - Primitive Roots

  1. #1
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    Primitive Roots

    Find a primitive root modulo 89 and use the answer to find a number a so that the order of a modulo 89 is 8.


    Does this mean

    a^8=1 (mod 89)?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Richmond View Post
    Find a primitive root modulo 89 and use the answer to find a number a so that the order of a modulo 89 is 8.


    Does this mean

    a^8=1 (mod 89)?
    if you are talking multiplicative order, then yes
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  3. #3
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    Quote Originally Posted by Richmond View Post
    Find a primitive root modulo 89 and use the answer to find a number a so that the order of a modulo 89 is 8.
    We use the result that the primes p>3 which have 3 as a quadradic residue are of the form 12k\pm 1. This result can be established using quadradic reciprocity. Since 89, a prime, does not have this form it means (3/89) = -1. Thus, 3^{(89-1)/2}= 3^{44} \equiv -1 (\bmod 89) by Euler's criterion. Let k be order of 3 then k|(89-1) \implies k|88. But if k=2,4,8,11,22,44 then k|44 and it would follow that 3^{44}\equiv 1(\bmod 89) which is a contradiction. Therefore, k=88 and consequently 3 is a primitive root of 89.

    It follows that 3^{11} \equiv 37 has order 8 mod 89.
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  4. #4
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    thank you.
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