Exactly $\frac{p-1}{2} - \phi(p-1)$ incongruent quadratic nonresidues mod p that are not primitive roots mod p

Question

Prove that there are exactly $$\frac{p-1}{2} - \phi(p-1)$$ incongruent quadratic nonresidues modulo $p$ that are not primitive roots modulo $p$.

I have been looking at this problem for quite some time, but have not been able to make any headway. Can you assist?

Isn't this just a matter of subtracting the number of primitive roots mod $p$ from the number of quadratic nonresidues? — hardmath, Nov 13 '13 at 03:44

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

1

HINT:

First of all, if $a$ is a Quadratic Residue $\pmod p,$ there exists $x$ such that $x^2\equiv a\pmod p\implies a^{\frac{p-1}2}=x^{p-1}\equiv1\pmod p$

$\implies$ ord $_pa|\frac{p-1}2\implies a$ can not be a primitive roots i.e., primitive root must be quadratic nonresidue.

Now use this and this

edited Apr 13 '17 at 12:20

Community

answered Nov 13 '13 at 03:47

lab bhattacharjee

Thank you! It's important to establish that primitive roots must be quadratic nonresidues. Thanks again. – Numbersandsoon Nov 13 '13 at 16:11
@BoSchmidt, I realized that when I was supplying the links:). My pleasure – lab bhattacharjee Nov 13 '13 at 16:12

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