Q1. How to find primitive root and generators modulo $97$?
Q2. Is there anyway to find the generators modulo $97$ in an easy way?
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For Q1, why not try the residue class $k = 2, \dots \frac{97 - 1}{2}$? If $k^{97 - 2} \ne 1 \bmod 97$, then $\operatorname{ord}_{\mathbb{Z}/(97 - 1)\mathbb{Z}} k = 97 - 1$, which is a primitive root / generator. – Alex Vong Nov 09 '17 at 09:53
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https://math.stackexchange.com/questions/124408/finding-a-primitive-root-of-a-prime-number – lab bhattacharjee Nov 09 '17 at 09:54
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@AlexVong That's not right. $2^{95}\neq 1$, but the order of $2$ is $48$ (in fact, any element with an even order would satisfy $k^{97-2}\neq 1$). – Arthur Nov 09 '17 at 09:55
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What is the difference between primitive roots and generators? – lhf Nov 09 '17 at 09:58
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@Arthur You are right, we should instead find the factors of $97 - 1$, called them $a_i$, and check $k^{a_i} \ne 1$. – Alex Vong Nov 09 '17 at 10:06
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@lhf I think they are just different names. – Alex Vong Nov 09 '17 at 10:10
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hi sir i dont get it could u add a site Or book that should help me out in this i would be very thank ful – hassampir pir Nov 09 '17 at 10:12
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1https://en.wikipedia.org/wiki/Primitive_root_modulo_n#Finding_primitive_roots – Alex Vong Nov 09 '17 at 10:15
2 Answers
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No efficient method is known for finding primitive roots, though typically there is a small one.
For $97$, the smallest primitive root is $5$. You need to test $2$ and $3$ but not $4$ since $2$ fails.
Once you find a primitive root mod $p$, call it $g$, then all other ones are $g^k$ with $k$ coprime with $p-1$.
lhf
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HINT
First notice that $97$ is prime.
Now prove-or take for granted if you must-that $x\neq 0$ is a primitive element modulo $97$ iff $x^{32}\equiv 1\operatorname{mod} 97$ and $x^{48} \equiv 1\operatorname{mod}97.$
MathematicianByMistake
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