Questions concerning taking n'th roots in modular arithmetic
Questions tagged [modular-nth-roots]
6 questions
4
votes
2 answers
Solve $x^{5} \equiv 2$ mod $221\ $ [Taking modular $k$'th roots if unique]
We know that $221 = 17*13$. So we can check if the system has roots to both of those equations separately, which it does:
$x^{5} \equiv 2$ mod $13$ has the solution $6 + 13n$ and $x^{5} \equiv 2$ mod $17$ has the solution $15 + 17n$.
I got these…
Wallace
- 436
3
votes
2 answers
Is there any way to predict the largest number of consecutive quadratic or cubic residues modulo prime $p$?
We all know that $a$ is a quadratic residue modulo $p$ if and only if $a^{(p-1)/2} \equiv 1 \pmod p$,
also $a$ is a cubic residue modulo $p$ if and only if $a^{(p-1)/3} \equiv 1 \pmod p$.
Now, for a given prime $p$:
(1) is there any way to predict…
2
votes
1 answer
Is there a formula for $|H_n|$, where $H_n = \{ $ units $u \pmod n$ such that $u^n = u, \}$ is the group of $(n-1)$th roots of unity modulo $n$?
Denote the group of solutions $X$ modulo $n$ to
$$
X^{m} = X \pmod n
$$
by $H(m,n)$. Then $H(m,n)$ is a subgroup of $G_n = \Bbb{Z}_{n}^{\times}$ the group of units modulo $n$. Note that $H(n-1,n) = G_n$ and $H(n+1, n+2) = G_{n+2}$ if and only if…
Daniel Donnelly
- 22,288
2
votes
1 answer
Writing integers modulo p as the sum of a quadratic residue and quadratic non-residue
I am interested if for a given $x \in\mathbb Z/p\mathbb Z$ we can write $$x = a+ b \tag{$*$}$$ for $a$ a quadratic residue and $b$ a non-residue modulo $p$.
If $p \equiv 3 \pmod 4$ this is true since we can write always write $x = r^2 - s^2$ as the…
Red Grassrex
- 181
1
vote
2 answers
Uniqueness of $n$'th roots $\bmod p\,$ when $n$ is coprime to $p-1$
Prove, that if $n$ is coprime to $p - 1$ ($p$ is prime), then exponentiation to the $n$th power in the residue field modulo $p$ is a bijection.
I tried proving that $a^n \equiv b^n \pmod{p} \Rightarrow a = b$, but I couldn't.
Also I have to prove…
user678243
0
votes
1 answer
Does Knowing Two Proper Factors of an RSA Public Key Permit Decryption if $N$ is Not a Semiprime?
It is my understanding that the public key modulus $N$ for the RSA cryptosystem is presumed to be a semiprime. I have also read where it is not necessary that $N$ be a semiprime, but it could be some other type of composite number.
My question is:…
DDS
- 3,289