Questions tagged [primitive-roots]

For questions related to primitive roots, the generators of the multiplicative group of integers modulo n.

Filter by
Sorted by
Tagged with
-1
votes
0answers
89 views

Help with some elements of GF(17^2) [closed]

I am stuck with the following: I want to find some elements $a$ and $b$ of ${\bf GF}(17^2)$ such that. $a^9=1$ and $a^3=16$. I have no access to some computational program to assist. Any assistance?
1
vote
0answers
142 views

A question related to Artin's Conjecture

I have the following question: Given an integer $n \ge 1 $ which is not a square, does there exist a prime number $p$ for which $n$ is a primitive root modulo $p?$ It is easy to see that if $n$ is a ...
5
votes
1answer
291 views

The smallest primitive root modulo powers of prime

I wrote a program to calculate the minimal primitive root modulo $p^a$ where $p > 2$ is a prime, by enumerating $g$ from $2$ and checking whether it's a primitive root, but I forgot to check $\gcd(...
5
votes
2answers
228 views

Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?

Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the ...
9
votes
3answers
541 views

Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$?

As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$? I have been thinking about this ...
4
votes
0answers
104 views

Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...
6
votes
1answer
229 views

Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?

Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$. Obviously, $\sigma_g(p)=\begin{pmatrix} 1 & 2 & \ldots & {p-1} \\ g^1\pmod{p} & g^2\pmod{p} &...
3
votes
1answer
719 views

list of primes for which 2 is a primitive root

I am looking for a list of primes $\le N$ for which 2 is a primitive root, with $N$ as large as available. I know of a table of smallest primitive roots for all primes below 1000, from which such a ...
2
votes
1answer
194 views

How to prove an approximation of a combinatorics identity

How to prove that $$\sum_{k\ge 0} \binom{n}{rk} =\frac{1}{r}\sum_{j=0}^{r-1}(1+w^j)^n$$ can be approximated as $\frac{2^n}{r}$, where $n\ge 0$, $r\ge 0$, $n>r$ and $w^r=1$ is a primitive $r$-th ...
2
votes
1answer
118 views

Bases of the special form

Let $R = \mathrm{GF}(q)$, $S = \mathrm{GF}(q^n), \ n\geq 2$ be extension of $R$, $h$ be a primitive element of $S$. I want to count or estimate the number $N$ of bases of the following form. Let $$\...
6
votes
0answers
513 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
1answer
424 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
2answers
506 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
1answer
224 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 $\...
9
votes
1answer
1k 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 $G(...
3
votes
1answer
433 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 ...
10
votes
2answers
716 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 ...
10
votes
2answers
1k 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 $p$.
22
votes
3answers
947 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 ...
15
votes
4answers
3k 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 ...
2
votes
3answers
912 views

Generalization of primitive roots

The standard definition is that $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the order of $a$ modulo $p$ is $p-1$. Let me rephrase, to motivate my generalization: $a\in\mathbb{Z}$ is a ...