# 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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### Understanding steps of calculating discriminants.

Here is the question I am asking about some steps in its keen answer: For $n =3,$ write $\Delta^2$ as an element of $A = \mathbb{Q}[e_{1}, e_{2}, e_{3}.]$(manually) The answer is given below: What a ...
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### Do primitive roots mod m always satisfy $\gcd(r^t,m)=1$?

I am confused about primitive roots. My text defines primitive roots as the solutions for $a$ of the equation $\operatorname{ord}_m a = \phi(m)$ where $\operatorname{ord}_m a$ is defined as the ...
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### If r is a primitive root mod m, then r is a primitive root $\pmod{\phi(m)}$?

Gerstein's Introduction to Mathematical Structures and Proofs offers the following proposition and corollary: Suppose r is a primitive root mod m: Prop 6.80: $log_r xy \equiv log_r x + log_r y$ ...
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### If r is a primitive root, then the residue of $r^t$ is also a primitive root if $\gcd(t,\phi(m))=1$ where $\phi$ is Euler's totient

This is part ii of the proof of Proposition 6.77 of Gerstein's Introduction to Mathematical Structures and Proofs. I don't understand it. Here is how the discussion, and my understanding of it, go: $r$...
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### Order of 2 modulo p, where p is a prime divisor of the Fermat number $F_n=2^{2^n}+1$

The order of 2 modulo p is the minimal solution of $2^t\equiv 1 \pmod{p}$ Euler's theorem guarantees that the congruence has a solution. The challenge is to demonstrate that $k=2^{n+1}$ is the minimal ...
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### If $p$ is an odd prime and $\alpha\in\Bbb Z/p\Bbb Z^*$, then $\alpha^2$ is not a primitive root modulo $p$.

Prove true or give a counterexample if false. If $p$ is an odd prime and $\alpha\in\Bbb Z/p\Bbb Z^*$, then $\alpha^2$ is not a primitive root modulo $p$. I was trying to prove it to be true, but I ...
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### How to show that 2 is primitive root modulo $101^{101}$?

Given the facts: (a) 2 is primitive root modulo 101 (b) $2^{50}+1$ isn't divisible by $101^{2}$ I have been asked to show that 2 is a primitive root modulo $101^{101}$. How do I do that? I started by ...
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