# 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.

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### If $p$ is an odd prime with primitive root $1<r<p$, is $r$ also a primitive root modulo $p^2$?

I use excel computed till $p=23$, it's true. But is this always true? if not, could you pls give a counter example?
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### Relationships between the Orders of an element in a Cyclic group (multiplicative and additive) Relating to prime numbers

Let p be a prime. Then, we know that $U(\mathbb{F}_p) \simeq \mathbb{Z}/(p-1)\mathbb{Z}$, where $U(\mathbb{F}_p)$ is the group of units of the field $\mathbb{F}_p$. They both have order $p-1$. Given ...
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### Principal vs primitive $n$-th roots of unity [closed]

Can someone please explain the difference between principal and primitive $n$-th roots of unity ? I know what $n$-th root of unity is but don't seem to understand these two concepts. Can someone ...
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### Proving two distinct primitive roots do not generate $\mathbb{Z}^{\times}_n$ in the same order

For any suitable $n$ that has primitive roots (i.e. $n$ of the form $2, 4, p^j, 2p^j$, where $p$ is an odd prime), there exist primitive root(s). In the case that $n$ has more than one primitive root, ...
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### Generalization of Artin conjecture to square of primes [closed]

Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. Can we expect that it is also a ...
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### Prove that $7$ is a primitive root modulo $p=2^s+1$. [duplicate]

I need help with the following question: If $1<s\in \mathbb N$ and $p=2^s+1$ is prime, so $7$ is a primitive root modulo $p$. My thoughts: First I know: $\phi(p)=p-1=2^s$. So if $r$ is the order ...
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### Quadratic number fields that contain primitive root of unity

Find all quadratic fields $\mathbb{Q}[\sqrt{d}]$ that contain some $p$-th primitive root of unity, where $p>2$ is a prime. Now, my reasoning was: if $\mathbb{Q}[\sqrt{d}]$ contains one $p$-th root ...
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### Is there a deeper reason for the classification of moduli in which a primitive root exists?

The primitive root theorem classifies the set of moduli for which a primitive root exists as $$1,2,4,p^k,2p^k$$ where $p$ is an odd prime and $k$ is a positive integer. I have worked through a proof ...
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### For every $k \in \Bbb Z$ there is $0 \le x \le p-1$ such as $x^3\equiv k \pmod {p}$

This question is looking like an easy one but I have been trying to solve it for the last couple days and I haven't been able to prove it - so I need some help. The question: Let $p$ be a prime number,...
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### coprime terms in the factorization of $x^p + y^p$

This is a question related to Fermat's last theorem. Let $p\geq5$ be a prime number, and let $\zeta$ be a primitive $p$th root of unity. Consider the Fermat's last theorem: \begin{equation} z^p = (x+y)...
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### Primitive root modulo $2p$

The question: Let $a,p \in \Bbb N$,$\$ $p$ is an odd prime, $a$ is a primitive root modulo $p$. prove that: if $a$ is odd, $a$ is primitive root modulo $2p$. if $a$ is even, $a+p$ is primitive root ...
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### Prove that $a$ is primitive root modulo $p^2$

I really need to answer this question quickly for my homework due tomorrow: Let $a,p \in \Bbb N$ $p$ is prime, $a$ is a primitive root modulo $p$ that $p^2\nmid (a^{p-1}-1)$. Prove that $a$ is ...