Questions tagged [primitive-roots]

For questions about primitive roots in modular arithmetic, index calculus, and applications in cryptography. For questions about primitive roots of unity, use the (roots-of-unity) tag instead.

395 questions
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Prove sum of primitive roots congruent to $\mu(p-1) \pmod{p}$

Suppose that $p$ is a prime. Prove that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \pmod{p}$.
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Are there infinitely many primes $n$ such that $\mathbb{Z}_n^*$ is generated by $\{ -1,2 \}$?

Let $n$ a prime, and let $\mathbb{Z}_n$ denote the integers modulo $n$. Let $\mathbb{Z}^*_n$ denote the multiplicative group of $\mathbb{Z}_n$ Are there infinitely many $n$ such that $\mathbb{Z}^*_n$ ...
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Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them

Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them. Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ ...
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Any element of $\mathbf{Z}[\xi]$ is congruent to an integer modulo $(1-\xi)^2$ if multiplied by a suitable power of $\xi$

I'm currently reading Kummer's famous paper on Fermat's Last Theorem (if anyone wants the link, I'll post it, but the paper is in German). There's the following statement in there, which should be "...
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What are primitive roots modulo n?

I'm trying to understand what primitive roots are for a given $\bmod\ n$. Wolfram's definition is as follows: A primitive root of a prime $p$ is an integer $g$ such that $g\ (\bmod\ p)$ has ...
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Is every non-square integer a primitive root modulo some odd prime?

This question often comes in my mind when doing exercices in elementary number theory: Is every non-square integer a primitive root modulo some odd prime? This would make many exercices much ...
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Number of primitive roots mod $p$ that are not primitive roots mod $p^2$

Consider the primitive roots of a prime $p$ in the range $1...p$ which are not primitive roots mod $p^2$. Let $n(p)$ be this number. While looking for an answer to this question, it seems that the ...
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AMM 2488: Primitive Root Relatively Prime to p-1

(from American Mathematical Monthly, problem 2488. I hope this hasn't been posted before but I'm new and maybe not very good at using the search function effectively..) Let $p>3$ be a prime. Show ...
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Find the last digit of the exponent $x$.

Let \begin{align} p&=396543857870745963499374527519378569849832249490600276007703072957912\cdots\\ &\phantom{=}8049490077183813353745228056691 \end{align} This number is a 100-digit prime ...
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Reluctant roots: $n$ is a primitive root of $p$ but not of $p^2$

I was looking at the primitive roots $n \bmod p$ and $p^2$ to see how often we get primitive roots of a prime that are not primitive roots of the square of that prime. I'll call this a reluctant root ...
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Show that $-3$ is a primitive root modulo $p=2q+1$

This was a question from an exam: Let $q \ge 5$ be a prime number and assume that $p=2q+1$ is also prime. Prove that $-3$ is a primitive root in $\mathbb{Z}_p$. I guess the solution goes something ...
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Question about primitive roots of p and $p^2$

If $g$ is a primitive root of a prime $p$, then $g$ is also a primitive root of $p^2$ if and only if $g^{p-1} \pmod p^2$ is not $1$. Is there a prime $p$ such that $p^2$ missing exactly $m$ primitive ...
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2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
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primitive roots problem. that integer n can never have exactly 26 primitive roots.

Show that no integer $n$ can have exactly 26 primitive roots. I know that if $n$ has primitive roots then it has exactly $\phi(\phi(n))$ primitive roots. I think the proof has to use contradiction. ...
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Solve $7^x \equiv 6 \pmod{17}$ given 3 is a primitive root $\bmod 17$

It's easy to show that 3 is a primitive root $\bmod 17$, but how do I use it prove the congruence? Is there a general way to solve any congruence of the form $a^x \equiv b \pmod{c}$ if you know a ...
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integer $m$ has primitive root if and only if the only solutions of the congruence $x^{2} \equiv 1 \pmod m$ are $x \equiv \pm 1\pmod m$.

Show that the integer $m$ has primitive root if and only if the only solutions of the congruence $x^{2} \equiv 1 \pmod m$ are $x \equiv \pm 1\pmod m$. I don't quite understand what this question is ...
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Prove that $\eta - \omega \notin \mathbb{Q}$ where $\omega$ and $\eta$ are two differents n-th primitive roots $\in \mathbb{C}$

Let $n \in \mathbb{N}$ be a natural number, and be $\omega$ and $\eta$ two differents n-th primitive roots in $\mathbb{C}$. Prove that $\eta - \omega \notin \mathbb{Q}$ My attempt was to follow the ...
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Prove that $\gamma^{\frac{q-1}{2}}=-1$ in $\mathbb{F}_q$ when $q$ is an odd prime power and $\mathbb{F}_q^*=<\gamma >$.

I have written a proof for this, however I think there may be a simpler way to go about it and am curious for any suggestions. Also I don't fully see where my proof would fail if $q$ was even (other ...
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Primitive roots modulo primes congruent to n!

for $N \ge 4$. Show for prime numbers, $p \equiv 1$ mod $(N!)$ that none of the numbers $1,2,...,N$ are primitive roots modulo $p$ I can't figure out where to start with this question, all I can ...
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A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.

Let $p=2^{2^k}+1$ be a prime where $k\ge1$. Prove that the set of quadratic non-residues mod $p$ is the same as the set of primitive roots mod $p$. Use this to show that $7$ is a primitive root mod $p$...
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Every primitive root modulo an odd prime is a quadratic nonresidue

This is my proof of the title statement. Is it correct? Suppose $a$ is a primitive root and quadratic residue modulo $p$. Then by definition $$\operatorname{ord}_p(a)=p-1$$ But Euler's ...
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Primitive root pairs

Looking at the Wikipedia entry for Primitive roots modulo $n$, I noticed that for some $n$, they came in pairs: When $n=p^k$ or $2p^k$ and $p\equiv1 \mod{4}$, then $r$ is a primitive root iff $n-r$ ...
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Solve $x^2 = 2$ over $F_5$.

Since $F_5$ is isomorphic to $\Bbb Z_5$, I tried to solve this equation over $\Bbb Z_5$. Since $gcd(2,5)=1$, $\Bbb Z_5$ contains a primitive $2$nd root of unity. So if $\omega$ is the ...
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No primitive root modulo $2^n$ for $n\ge 3$

Prove that there is no primitive root modulo $2^n$. I'm not sure how to begin proving this. I know $\varphi(2^n)=2^{n-1}$, thus a primitive root $a\in\left(\dfrac{\mathbb{Z}}{2^n\mathbb{Z}}\right)^*$ ...
Find all the solutions of $y^2 \equiv 5x^3 \pmod 7$
I need to find all the solutions of $y^2 \equiv 5x^3 \pmod 7$. I managed to solve by trying one-by-one, but I guess there is some other way to solve this.