Questions tagged [multiplicative-order]
Let $G$ be a finite group, typically $\mathbb{Z}/n \mathbb{Z}$, and $g\in G$. The multiplicative order of $g$ is the least $n\in\mathbb{N}^+$ such that $g^n = e$, the identity of $G$.
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What is the order of $p_{1}^{x} \bmod{n}$ where $p_1$ is a prime factor of $n$ [closed]
I am looking for a formula, algorithm, or even literature on the topic.
Take $21$ for example
$21 = 7 \cdot 3$
What is the order of $3^{x} \bmod 21$?
$3^0 = 1$
$3^1 = 3$
$3^2 = 9$
$3^3 = 6$
$3^4 = 18$...
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Properties of matrices with a multiplicative order
I have been trying to find any article or sources talking about the structure and properties of matrices with a multiplicativw order, i.e.
a matrix $A$ has a multiplicative order of $n$ if and only if ...
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Infinite sequence of integers $\{m_n\}$ for which the orders of both 2 and 3 are small modulo $m_n$.
It is easy to create a sequence $\{m_n\}$ for which the order of $2\pmod{m_n}$ is as small as possible, i.e. it is about $\log_2(m_n)$. For example $m_n=2^n-1$ is an appropriate sequence. But if I ...
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n = pq where p and q are odd prime numbers, gcd(c,n) = 1, c < n, show at most $\frac{��(n)}{4}$ of c satisfy $ord_n(c)$ is odd
Suppose n = pq where p and q are distinct odd prime numbers. Show that, out of the φ(n) different integers c satisfying 1 < c < n and gcd(c,n) = 1, at most $\frac{φ(n)}{4}$ of them have
the ...
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Why is the multiplicative order of perfect powers modulo $p$ smaller on average?
I've been trying to analyse by myself for recreational purposes what would be a "better" base to use instead of the common decimal one. Part of what should make a base better is to have ...
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$m+1$ generates the kernel of $Z_n^\times\to Z_m^\times$ where $m\mid n$ with the same prime factors
Suppose $m\mid n$. Using the First Isomorphism Theorem with respect to the homomorphism $$\begin{array}{rccc}f:&\mathbb{Z}_n^\times&\to&\mathbb{Z}_m^\times \\&x&\mapsto &x\bmod ...
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About order of a number modulo 2. [duplicate]
For $a, m, n \in \mathbb{N}$, prove that $$\gcd(a^{2^m} + 1, a^{2^n} + 1) = \begin{cases} 1, & \text{if $a$ is even}\\ 2, & \text{if $a$ is odd} \end{cases}$$
given $m \ne n$.
My attempt:
...
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Multiplicative order of $2$ modulo $p$.
When calculating the multiplicative order of $2$ modulo a prime $p$ you often get $p-1$ or $\frac{p-1}{2}$ as a result, but there are cases where this does not hold, is there a general form for those ...
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When $\text{ord}_n(p)/m = \text{ord}_{n/m}(p)=1$
$\newcommand\ord{\text{ord}}$
For integers $n,m,p$, suppose $m$ and $n$ share the same prime divisors with $m$ dividing $n$, and suppose $\text{gcd}(p,n)=1$.
I want to show that $\ord_n(p)/m=\ord_{n/m}...
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Roots of unity over finite fields
Let $H = \{\omega, \omega^2, \dots, \omega^{n-1}, \omega^n = 1\}$ be the multiplicative subgroup of $\mathbb{Z}_p$ of $n$-th roots of unity, generated by the primitive $n$-th root of unity $\omega$.
I ...
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Multiplicative Order when for integers $m > 2, n > 0$, when $2^m | (3^n - 1)$
It seems to me that for integers $m>2, n>0$, when $2^m | (3^n - 1)$, that the multiplicative order is $2^{m-2}$ so that for $0 < i < 2^{m-2}$, $3^i \not\equiv 1 \pmod {2^m}$ and $3^{2^{m-2}...
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What is the order of $\bar{2}$ in the multiplicative group $\mathbb Z_{289}^×$?
What is the order of $\bar{2}$ in the multiplicative group $\mathbb Z_{289}^×$?
I know that $289 = 17 \times 17$
so would it be $2^8\equiv 256\bmod17 =1$
and therefore the order of $\bar{2}$ is $8$? I'...
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For which $n$ does $p_n(x)=\sum\limits_{{k=1,(k,n)=1}}^n o(k) x^k $ have exactly two real roots?
Let $n\in\mathbb{N}$ be fixed and denote by $o(k)$ the multiplicative order of $k$ modulo $n$. Define $$p_n(x)=\sum_{\substack{k=1 \\ (k,n)=1}}^n o(k) x^k ;$$here the sum is taken over $k$ that are ...
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What is the order of $\bar{2}$ in the multiplicative group $\mathbb{Z}_{221}^\times$?
What is the order of $\bar{2}$ in the multiplicative group $\mathbb{Z}_{221}^\times$?
I keep computing $2^0, 2^1,...$ until we get $1\ (\operatorname{mod} 221)$, but this will take forever! Is there ...
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What is the order of the multiplicative group?
According to Lagrange's Theorem, the multiplicative group $(\mathbb{Z}_{54})^\times$ cannot contain a subgroup of which order:
A: $9$
B: $18$
C: $6$
D: $12$
I think that it is D because $12$ is not a ...
user828643