# 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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### 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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### 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 '...
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### I've followed two different methods to find congruence with primitive roots but have two different answers

So I'm using the fact that 2 is a primitive root modulo 53, I'm solving $x^5 \equiv 8\mod{53}$ Originally I was trying to rewrite both sides in terms of two so had the following: let $x=2^y$ for y ...
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I've read through this answer to get some ideas: Solving a congruence using a primitive root But my problem is slightly different and it's thrown me off in terms of understanding the logic. I have $x^... 1 vote 0 answers 186 views ### For what primes is$6$a primitive root? While thinking about an unrelated problem, I had to decide whether$6$was a primitive root with respect to multiple prime moduli. I could discover no obvious pattern as to primes for which$6$is a ... • 7,523 1 vote 0 answers 69 views ### Solving the equivalent congruence I found and using it to derive the solution to the original congruence (elementary number theory) I am trying to solve following problem. I have done the entire problem, so I'm not asking anyone to do the problem for me. But I need some confirmation on whether or not the very last part of my ... 1 vote 0 answers 83 views ### Some problems on primitive roots and divisors in number theory. I have two questions: Let$n>1$be a positive integer that isn't a perfect power. Is$n$a generator mod infinitely many primes? Let$C,x,y$be pairwise coprime integers$\in \mathbb{N}$. Let$...
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My question is about the roots of unity in finite fields. It goes like this: Suppose we have two primes p and q, both greater than 3, which satisfy $q|(p-1)$. Then there exists a $q$-the root of unity ...