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I use excel computed till $p=23$, it's true. But is this always true? if not, could you pls give a counter example?

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  • $\begingroup$ That's not very far. $\endgroup$ – Angina Seng Sep 20 at 10:20
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    $\begingroup$ It may happen that $r^{p-1}\equiv 1\pmod{p^2}$, I suppose $\endgroup$ – Hagen von Eitzen Sep 20 at 10:30
  • $\begingroup$ @AnginaSeng thanks for the hint, yeah I just need to go one step further :p $\endgroup$ – athos Sep 20 at 10:34
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    $\begingroup$ Try Table[SubsetQ[PrimitiveRootList[Prime[n]^2], PrimitiveRootList[Prime[n]]], {n, 1,100}] on Mathematica. $\endgroup$ – Chrystomath Sep 20 at 10:43
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    $\begingroup$ See also math.stackexchange.com/a/1599492/589 and oeis.org/A060503 $\endgroup$ – lhf Sep 20 at 10:52
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just to dot it that $r=14$ is a primitive root for $p=29$, but not for $p^2=841$. Instead $r+p = 43$ is a primitive root for $841$.

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