The primitive-roots tag has no wiki summary.

**5**

votes

**0**answers

236 views

### Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...

**3**

votes

**1**answer

275 views

### Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime ...

**8**

votes

**2**answers

395 views

### Proof theory and primitive roots

I have had this question on my mind for two decades. We know, after Heath-Brown, that one out (say) of 3, 5, 7 is a primitive root mod p for infinitely many primes p. We just don't know which one. (We ...

**0**

votes

**1**answer

210 views

### Order of difference of two generators of cyclic group

Let $n\in\mathbb{Z}^+$ and $\alpha,\beta $ be two generators of the cyclic group $\left(\frac{\mathbb{Z}}{(2^n - 1)\mathbb{Z}},+\right)$.
Question: What are known theorems regarding the order of ...

**7**

votes

**1**answer

592 views

### Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* .
Is it known that ...

**3**

votes

**1**answer

280 views

### primitive root 2 in (Z/pZ)* for prime p and generating GF(2^{p-1})

It appears (from computer experiments) that if $p$ is a prime such that 2 generates the multiplicative group $\mathbb{F}_p^\times$ of the corresponding finite field $\mathbb{F}_p$, then the ...

**9**

votes

**2**answers

592 views

### Approximate primitive roots mod p

Artin conjectured that if $a$ is an integer which is not a square and not $-1$ then $a$ is a primitive root for infinitely many primes. This conjecture has not been resolved, but partial results are ...

**8**

votes

**2**answers

917 views

### primitive roots and primes

Given a positive integer $n > 1$, is it true that there exists infinitely many primes $p$ such that $n$ is a primitive root modulo $p$.

**21**

votes

**3**answers

805 views

### Primitive roots

If Artin's conjecture on primitive roots is true, then 2 generates $(\mathbb{Z}/p\mathbb{Z})^\times$ for infinitely many primes $p$. Can one at least show that $(\mathbb{Z}/p\mathbb{Z})^\times$ is ...

**10**

votes

**4**answers

2k views

### Is the smallest primitive root modulo p a primitive root modulo p^2?

Let $p \ne 2$ be a prime and $a$ the smallest positive integer that is a primitive root modulo $p$. Is $a$ necessarily a primitive root modulo $p^2$ (and hence modulo all powers of $p$)? I checked ...