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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.

582 questions
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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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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 roots of $p$ , for arbitrarily large $m$ ? Does…
sumpeloh miyapah
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Proving a number has no primitive roots

How do you prove an arbitrary number $n$ has no primitive roots without finding all numbers less than $n$ which are also coprime to $n$ and exhausting that none of the order of these numbers modulo $n$ are equal to $\phi(n)$?
ddresner8
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About primitive roots and primes.

For any odd prime $p$ there exists at least one prime $q < p$ such that $q$ is a primitive root $\text{mod } p$ ; is this true?
201044
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Primitive Roots modulo p

I'm asked the following question: Prove that $b$ is a primitive root modulo $p \implies$ the smallest positive exponent $e$ such that $b^e \equiv 1 \pmod p$ is $p - 1$. I know that this could probably be shown easily with Fermat's Little Theorem,…
Brandon
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Primitive Root Modulo $m$

I need help with the following: Show that if $b$ is a primitive root modulo $m$, then $$\{b,b^2,b^3,...,b^m-1\}$$ is a complete set of units modulo $m$.
Benji_Bombadill
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Prove that number is a primitive root for all $k$ in range

Let $a$ be a primitive root for $p > 2$ where $p$ is prime. Show that $a^p+kp$ for $k = 1,\dots,p-1$ are $p-1$ distinct primitive roots modulo $p^2$. What I have done: First of all it is easy to see that by Fermat' Little Theorem $a^p+kp \equiv a +…
Kristin Petersel
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Why the result is not $0$?

Applying some properties, this should be: $(\sqrt[n]{a} * \sqrt[k]{a} ) - (a^{\frac{n+k}{nk}})= 0$ But according to symbolab, not. I guess it's a symbolab error. But I would like to make sure.
ESCM
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Primitive Root mod 26 and 25?

I would live to calculate the primitive roots modulo 26 and modulo 25. My approach: 26 is not a prime number. But 26=2*13 are Prime numbers. So I calculated the primitive roots of them: Result for 2: 1 Result for 13: 2,6,7,11 So what are the…
NumberTheory
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How do I primitive $\sin(x)\cos^3(x)$ step by step?

I tried to primitive $\sin(x)\cos^3(x)$ step be step, but I got stuck. Can I use substitution? $[\sin(x)]' = \cos(x)$ And write $\cos^3$ as $\cos^2(x)*\cos(x)$ And also write the function $\sin(x)\cos^2(x)*\cos(x)dx$ as $\sin(x)\cos^2(x)d(\sin(x))$
Dorians
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[Proof]if p and q are odd prime, and q=2p+1,then -4 is primitive root of q

how to proof if p and q are odd prime, and q=2p+1, then -4 is primitive root of q. I think quadratic residue of q is useful, but I cant use it effectively. if for example. p=3, then q=7, and the number -4(mod 7) is the primitive of 7. my try is: if…
hwiba12
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$(e^{2\pi i}){}^n \neq e^{2\pi i n}$ where $n\in\mathbb{N}$?

When I type these equations into a calculator I get $({e^{2\pi i}}){}^n = 1$ and something else for $e^{2\pi i n}$. Is that due to the imprecision of the calculator or does the inequality follow logically?
jvdh
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$m$-th primitive root of unity over $\mathbb{Z}_{2^k}$ for some integer $k$

Can we find the $m$-th primitive root of unity over $\mathbb{Z}_{2^k}$ when $m$ is also a power of $2$?
mallea
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proof of primitive roots algorithm

We know the following from elementary number theory: (1) for $gcd(a,n) =1, a^{\phi(n)} \equiv 1 (mod \, n)$ (2) $ord_na \mid \phi(n)$ (3) $a$ is a primitive root $mod \, n$ iff $ord_na = \phi(n)$. From (2), the only possible values of $ord_ma$ are…
doctorjay
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