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It is well known that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \bmod{p}$.

But I can't see why this result is interesting or useful. Can someone please enlighten me?

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    Related: http://math.stackexchange.com/questions/25452/prove-sum-of-primitive-roots-congruent-to-mup-1-pmodp2012-01-12
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    @JavaMan, thanks, I've seen this and I have added my question there as a comment but I though an actual question would be more visible. Moreover, that question you mentioned does not touch my question.2012-01-12
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    After I included the link, I saw that you commented there. I wasn't sure whether to keep the link here or not, but I agree that asking this as a new question will attract more attention.2012-01-12
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    Is there a reason to think that it *is* particularly interesting or useful? I'm not saying it's not, but it looks like a homework problem relating a cute fact you get by looking at coefficients of minimal polynomials...2012-01-12
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    I think its main use is as a neat exercise you can give to a number theory class to see whether they figure out how to apply the concepts you've taught.2012-01-15
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    @Gerry, Wikipedia says this result was found by Gauss. He probably found it interesting. Perhaps I should look in *Disquisitiones Arithmeticae* (art. 81)...2012-01-24
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    It can never be a bad idea to look at the Disquisitiones.2012-01-25
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    A generalization of Gauss's results is given in G. A. Miller, [On the Sum of the Numbers Which Belong to a Fixed Exponent as Regards a Given Modulus](http://www.jstor.org/stable/2972422), *The American Mathematical Monthly*, Vol. 19, No. 3 (Mar., 1912), pp. 41-46. At the end of that paper, there are a few useful applications.2012-01-25
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    But Gauß had successfully applied the sum of primitive roots to prove some important facts of the theory of quadratic forms, further leading to the prototype of the theory of characters, right? I am not sure if this is correct, but it appears so...2012-02-08
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    @awllower, ah, if you can provide a reference, please add this observation as an answer. Thanks.2012-02-08
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    @Ihf: Surely, when I can again get my hands on that book, an answer should be within the options. But the veracity of this naïve idea still needs to be vouched.2012-02-08
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    @lhf One reason why it looks interesting to me is that $\mu (n)$ can be only three values: ${-1,0,1}$. Why should the sum of the primitive roots modulo their prime end up being only one of these three numbers? I would expect the sum to be scattered about the residue classes modulo that prime. However, that is only a partial answer. You also want to know why it is useful.2016-02-25

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