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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.

If $n$ is a positive integer, a primitive root modulo $n$ is an integer whose multiplicative order modulo $n$ is equal to $\varphi(n)$, Euler's totient function evaluated in $n$.
A primitive root modulo $n$ is often identified with its corresponding element of $\mathbb Z/n\mathbb Z$. With this identification, a number is a primitive root modulo $n$ if and only if it is a generator of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$, in which case this group is cyclic.
A primitive root modulo $n$ exists if and only if $n$ is equal to $2$, $4$, $p^k$ or $2p^k$ for some odd prime $p$ and some positive integer $k$.

For questions about primitive roots of unity, consider using the tag.

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Finding a primitive root of a prime number

How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
user27617
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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 $\mu(p − 1) \pmod{p}$.
Johan
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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 $\mathbb{Z}^*_n$ is generated by $\{ -1, 2 \}$? Artin's conjecture…
Matthew Kahle
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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 multiplicative order $p-1$ The main thing I'm confused…
user2200321
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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 easier. Unfortunately I seem unable to discover anything…
Bart Michels
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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 $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent…
Saurabh
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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 "very easy to prove": Let $\xi$ be a primitive…
Steven
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Proof of existence of primitive roots

In my book (Elementary Number Theory, Stillwell), exercise 3.9.1 asks to give an alternative proof of the existence of a primitive root for any prime. Let $p$ be prime, and consider the group $\mathbb{Z}/p\mathbb{Z}$. Suppose that the non-zero…
Noldorin
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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 number of primitive roots mod $p$ that are not…
Esteban Crespi
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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 that there exists a primitive root $r$ mod $p$ with…
James Li
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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…
Christmas Bunny
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Intuition behind this strange heuristic for primitive roots modulo $p$?

Let $p$ be an odd prime. Define $S(p)$ as the sum of all primitive roots modulo $p$ taken from $\left[-\frac{p-1}2,\frac{p-1}2\right]$. Now here's the strange thing. If the primitive roots were 'random', you'd expect $S(p)$ to be negative about as…
Mastrem
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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 of $p$. It's really rare. With $p$ limited to a…
Joffan
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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 criterion states that $$a^{(p-1)/2}\equiv1\bmod p$$ …
Lingnoi401
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$p^2$ misses 2 primitive roots

When I Checked primitive roots of some primes P, I found this following phenomenon: $14$ is a primitive root of prime $29$, but it's not primitive root of $29^2$ $18$ is a primitive root of prime $37$, but it's not primitive root of $37^2$ $19$…
pink tulips
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