# Tag Info

## Hot answers tagged primitive-roots

Accepted

### State of the art for primitive roots

Artin's conjecture on primitive roots has a qualitative version and quantitative version. The qualitative version says if $a \in \mathbf Z$ is not $-1$ or a perfect square then $a \bmod p$ is a ...
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### For a non-square, is there a prime number for which it is a primitive root?

We do not know that. It is typical for questions about the infinitude of primes satisfying some property that even proving the existence of at least one of those primes in sufficient generality is ...
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### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

Both Gauss' generalization, and the classification of moduli with primitive roots, are 'shadows' of the structural theory of the finite abelian group $G_m:=(\mathbb{Z}/m\mathbb{Z})^{\times}$. Gauss' ...
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### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

Expanding on a comment, the curve $X^4+Y^4=aZ^4$ (for $a\ne0$) has genus $3$. So the Hasse-Weil bound says $$N_p(a) := \#\bigl\{ [X,Y,Z]\in\mathbb P^2(\mathbb F_p) : X^4+Y^4=aZ^4 \bigr\}$$ satisfies ...
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### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

I will show the two results are non-superficially related by showing one of them implies the other: the classification of moduli $n \geq 2$ for which the unit group $(\mathbf Z/(n))^\times$ is cyclic ...
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### Arithmetic sequences and Artin's conjecture

It's plausible that there are infinitely many primes $p \equiv 1 \bmod 4$ of which $2$ is a primitive residue. However, this is false for $p \equiv \pm 1 \bmod 8$, because $2$ is a quadratic residue. ...
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### list of primes for which 2 is a primitive root

The sequence of primes for which 2 is a primitive root is OEIS A001122. It contains a list for the first 10.000 entries, which goes up to $N=310091$. I'm not sure if that is sufficient for you, but it'...
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### Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page ...
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1 vote
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### On a summation in "Artin's conjecture for primitive roots" by Heath-Brown

Consider the double sum on the right-hand side. If $n$ is a square, then $e+f+g+h$ is even, hence its contribution is asymptotically $\pi(x)$. If $n$ is not a square, then its contribution is \$o(\pi(x)...
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