Questions tagged [reduced-residue-system]
57 questions
6
votes
2 answers
Sum of residue systems is complete iff they are both complete
Let $p$ and $q$ be distinct prime numbers. The integers $a_1, \ldots, a_p$ and $b_1,\ldots,b_q$ are such that the sums $a_i + b_j$ form a complete residue system modulo $pq$ (that is, there is precisely one sum which is $0$ mod $pq$, precisely one…
DesmondMiles
- 2,931
6
votes
0 answers
Permutations of $\{1,\ldots,2pq\}$ modulo $2pq$
I am proposing here a variant of this problem.
Let $p$ and $q$ be distinct odd primes. Is it true that there exists a permutation $\sigma$ of $\{1,\ldots,2pq\}\times \{1,2\}$ such…
Paolo Leonetti
- 15,554
- 3
- 26
- 60
4
votes
1 answer
$\{r_1,r_2,...,r_{\phi(m)}\}$ is a reduced residue system modulo $m$ iff $\{r_1+k,r_2+k,...,r_{\phi(m)}+k\}$ be a reduced residue system modulo $m$
Let $2\lt m\in \mathbb N$ and $\{r_1,r_2,...,r_{\phi(m)}\}$ be a reduced residue system modulo $m$. I want to find a necessary and sufficient condition for $k$ such that the set $\{r_1+k,r_2+k,...,r_{\phi(m)}+k\}$ be a reduced residue system modulo…
hamid kamali
- 3,241
4
votes
0 answers
Question about the elements of a reduced residue system relative a primorial $p_n\#$
I've been dividing up the elements of reduced residue system relative a prime $p_n$ into congruence classes modulo $p_{n+1}$ and I noticed that each congruence class is represented.
If $r$ = the number of elements in a reduced residue system…
Larry Freeman
- 10,189
3
votes
1 answer
Question about congruence classes and reduced residue systems
Let $x$,$y$ be integers such that the reduced residue system modulo $y$ divides equally into congruence classes modulo $x$.
An example of this is $x=4$, $y=5$.
The reduced residue system modulo $5$ is $\{1,2,3,4\}$
These divide evenly into…
Larry Freeman
- 10,189
3
votes
1 answer
Estimating the number of elements with a given least prime factor in a sequence of consecutive integers
Let $a,n$ be any positive integers.
Let $\varphi(x)$ be the Euler totient function.
It seems to me that the number of elements $x$ with $a \le x < a+n$ that have a given least prime factor will be less than each of the following…
Larry Freeman
- 10,189
3
votes
0 answers
Reasoning about the number of elements in a reduced residue system relative a primorial
Let $R_{p_i\#}$ be the reduced residue system relative the primorial for the $i$th prime.
Let $\left|R_{p_i\#}\right|$ be the number of elements in this set.
It is well known that:
$$\left|R_{p_i\#}\right| =…
Larry Freeman
- 10,189
2
votes
1 answer
Reduced Residue System in Mathematica
How can I create the standard reduced residue system modulo $m$ in Mathematica for a given positive integer $m$? For example, if I input $10$, I would like it to give me the list $\{1,3,7,9\}$. Thanks.
Colin Defant
- 1,327
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2
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Solving for $x$ where $p^x \equiv 1 \pmod {q\#}$
For a given primorial $q\#$, you can generate a subset of the reduced residue system by using the power of a prime $p$ where $p > q$.
For example, for $5\#$, we can use the powers of $7$ to generate $4$ of the elements of the reduced residue…
Larry Freeman
- 10,189
2
votes
0 answers
Can all elements of a reduced residue class of a primorial $p$ be expressed as a simple equation in terms of the factors of the primorial?
I've noticed that for the smaller primes, it is possible to state each element of its reduced residue class as a simple equation in terms of the factors of the primorial.
For example, consider the primorial $5\# = 30$
The reduced residue class is…
Larry Freeman
- 10,189
2
votes
1 answer
Reasoning about reduced residue systems as a generalization from prime gaps
I have been thinking a lot about prime gaps.
It seemed to me that it is much easier to reason about reduced residue systems which can then be used to reason back to prime gaps.
Below is an example of thinking about prime gaps through reasoning about…
Larry Freeman
- 10,189
2
votes
0 answers
Prove that $Enc_{e}(\overline{a}) = \overline{a^e}$ is a one-to-one mapping
Task: Let $n \in \mathbb{N}, n \geq 2, e \in \mathbb{Z}, \left( e, \phi(n) \right) = 1$.
Prove that the mapping $Enc_{e}(\overline{a}) = \overline{a^e}$ mutually unambiguously maps (one-to-one mapping) the $\mathbb{Z}^{*}_{n}$ to itself.
Some…
Jacobs Monarch
- 460
2
votes
1 answer
Find all the natural numbers N such that the reduced residue system consisting of least positive residues modulo N form an arithmetic progression.
Firstly, I suppose that N is odd, then obviously the $1st$ term of the AP is $1$ and the $2nd$ term is $2$. Thus difference between $1st$ and $2nd$ term is $1$, thus the AP formed is $1, 2, 3, ... , N-1$. Thus $N$ must be prime. So we get that if…
2
votes
0 answers
Primes in reduced residue systems
I am trying to prove the following conjecture:
Let it be $R_m(n) / m>0, m\in \Bbb N$ some reduced residue system modulo $n$ such that $R_m(n)$ is the reduced residue system between $(m-1)n$ and $mn$, and such that some element of the reduced residue…
Juan Moreno
- 1,074
2
votes
1 answer
We have $a^{n-1} \equiv 1\ (mod\ n)$ but $a^{m} \not\equiv 1\ (mod\ n)$ for every divisor $m$ of $n - 1$, other than itself. Prove that $n$ is prime.
The integers $a$ and $n > 1$ satisfy $a^{n-1} \equiv 1\ (mod\ n)$ but $a^{m} \not\equiv 1\ (mod\ n)$ for every divisor $m$ of $n - 1$, other than itself. Prove that $n$ is prime.
The way I went about proving it was,
We know for every integer $k…
SS'
- 1,224