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Math Help - find primitive roots for mod

  1. #1
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    find primitive roots for mod

    is there an easy way to find primitive roots for a specific mod?
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  2. #2
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    There isn't a specific formula to find primitive roots. However, say that you have found one primitive root, g modulo m. Then, g^k is also a primitive root iff (\varphi (m), k) = 1 and this gives the complete list.

    More here: How to find a primitive root of N
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  3. #3
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    So say I have found a primitive root mod 29. namely 2.

    then the other primitive roots mod 29 are powers of 2?

    that means that 8 is also a primitive root because 2^3=8 and (28,3)=1.

    but 2^5=32 (28,5)=1 does that mean that 3 is a primitive root because 32 mod 29 is congruent to 3?

    is the only other primitive root for 29 8?
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    Yes. 2^k is a primitive root mod 29 iff (k,28) = 1.

    So for example, 2^3 \equiv 8 is a primitive root since (3, 28)=1
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  5. #5
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    but that is the only other p.r. besides 2. correct?
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  6. #6
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    No ... there are many other values for k such that (k,28) = 1.

    Try k = 1, 3, 5, 7, 9, 11, 13, ...

    In general, every prime p has \varphi (p-1) primitive roots.
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