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Tagged with primitive-roots number-theory
186
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Artin's conjecture on primitive roots for perfect powers
Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of ...
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Let p be a prime number with p≡3 (mod 4) and let r be a primitive root modulo p . Prove that $\mathrm{ord}_p(-r) = (p-1)/2.$
I only could write this:
Let p = 4k + 3 where k is an nonnegative integer.
Since r is a primitive root modulo p .
$r^{(p-1)/2} ≡ - 1 $ mod p. So
$r^{2k+1}≡ -1$ mod p
$(-r)^{2k+1}=-1*(r)^{2k+1}$
$-1*(...
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1
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tetration primitive root $q \mod p$
Consider primitive roots $q \mod p$ where $q$ is a prime and $p$ is an odd prime $> 5$.
I am looking for such pairs $q,p$ such that every residue $a_i \mod p$ is of the form
$$a_i = q^{(v_i)} \mod ...
- 16.4k
1
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1
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Show that $2$ is not a primitive root of $8k + 7$
I'm attempting to show that $2$ is not a primitive root of primes of the form $p = 8k + 7$. I know that, to do so, I must show that $2$ has order less than $\phi(p)$ modulo $p$ (where $\phi$ denotes ...
- 1,381
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1
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How to prove that $\sum_\limits{k=1\\(k,p-1)=1}^{p-1}g^k \equiv \mu(p-1)$ (mod p) for prime p and primitive root g
p is a prime and g is a primitive root modules p, and I want ot prove that:
$\sum_\limits{k=1\\(k,p-1)=1}^{p-1}g^k \equiv \mu(p-1)$ (mod p)
$\mu(x)$ is the Möbius function
I know how to deal with $\...
- 425
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Prove that the roots of cyclotomic polynomial $\Phi_{p-1}(x) \equiv 0 (mod~p)$ are exactly the primitive roots mod p
$p$ is a prime, and $\Phi_{p-1}(x)$ denote the cyclotomic polynomial of order $p-1$. And I want to show the following:
$g$ is a solution of the congruence $\Phi_{p-1}(x) \equiv 0 (mod~p)$ if and only ...
- 425
1
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1
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952
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A primitive root modulo p is a primitive root modulo $p^2$ if and only if $g^{p-1} \not\equiv 1 \mod{p^2}$
$p$ is an odd prime. I'm starting with number theory and I'm completly stuck with this question. In general, I don't really know how to approach the proves. Then I'm also supposed to prove that either ...
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How to solve the equation in algebraic number theory?
First step: When $p\equiv 1 \pmod{ 3}$, prove that there exists a pair $(a,b)$ of integers such that $4p=a^2+27b^2$, $a\equiv 1 \pmod{ 3}$ and a is unique (the proof of the first step).
Second step: ...
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2
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1
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About The Order of an Integer
In this bolg It says
$x=ord_{n}b$ and $ord_n = $ the least positive integer x such that $b^x\equiv $ 1 (mod n)
and below it says $b^x\equiv $ 1 (mod n) if and only if $ord_{n}b$ | x and then it ...
0
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1
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Showing polynomial $p(x) = x^2 −3 \in\mathbb{F}_7[x]$ is prime in $E[x]$ and $x^3 - 2\in\mathbb{F}_7[x]$ factors into linear terms in $E$
Here we define the set of equivalence classes $E[x] = \mathbb{F}_7[x]/(x^3 - 2)\mathbb{F}_7[x]$. I'm not sure if showing $p(x)$ is prime is equivalent to showing that it is a primitive root of $E^\...
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Number of roots in cyclotomic polynomial $\Phi_{15}[x]$ in $\mathbb F_p$
I'm trying to understand why if the $gcd(p-1, 15) = d \neq 15$, then there are zero roots (since if it's $=15$, there are exactly 8). I was thinking that since a solution to $x^d - 1$ is relevant if $...
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Find the sum of the $m$-th powers of the primitive roots mod p for a given prime p and a positive integer $ m$.
Wikipedia has the result that Gauss proved that for a prime number p the sum of its primitive roots is congruent to $\mu(p−1)\pmod p$ in Article 81.
also see:Prove sum of primitive roots congruent to $...
- 93.6k
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Are primes of the form $6k+1$ a cube modulo $n$, if $3\nmid n$ and none of the prime factors of $n$ is of the form $6k+ 1$?
I wonder if we can assume the following statement to be true in general:
Let $p$ be a prime of the form $6k+1$ and $n<p$ a natural number less than $p$. If $3$ does not divide $n$ and none of the ...
- 1,594
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How to solve the congruence $x^{30} ≡ 81x^6 \pmod{269}$ using primitive roots(without indices)?
So I know that 3 is a primitive root of 269.
How can I solve $x^{30} ≡ 81x^6 \pmod{269}$
Even if I substitute $x$ with $3^y$, where $y$ lies between 0 and 267, I can’t get any solutions.
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A primitive root modulo $p^k$ is primitive modulo $p^{k+1}$,for $k\geq 2$.
I am a graduate student of Mathematics.I am stuck with the following number theory problem:
Let $p$ be an odd prime.Prove that any primitive root modulo $p^k$ is a primitive root modulo $p^{k+1}$, for ...
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