All Questions
16 questions
37
votes
2
answers
3k
views
A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself
Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
- 1,730
19
votes
1
answer
3k
views
A mysterious connection between primes and squares
Motivated by two previous questions of mine (cf. Primes arising from permutations and Primes arising from permutations (II)), here I ask a curious question which connects primes with squares.
...
- 15.6k
10
votes
1
answer
694
views
Prime numbers from permutation
Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
- 4,938
8
votes
1
answer
259
views
For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)
Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...
- 1,871
7
votes
1
answer
531
views
Primes arising from permutations
Recently, Paul Bradley proved in arXiv:1809.01012 that for any positive integer $n$ there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that $k+\pi_n(k)$ is prime for every $k=1,\ldots,n$ (cf. ...
- 15.6k
7
votes
1
answer
304
views
Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?
Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$.
Obviously, $\sigma_g(p)=\begin{pmatrix}
1 & 2 & \ldots & {p-1} \\
g^1\pmod{p} & g^2\pmod{p} &...
- 3,866
6
votes
1
answer
367
views
Is $|\{(j,k):\ 1\le j<k\le\frac{p-1}2:\ \&\ (j^{16}\ \text{mod}\ p)>(k^{16}\ \text{mod}\ p)\}|$ even for each prime $p\equiv1\pmod {16}$?
In my paper http://arxiv.org/abs/1809.07766, I determined the parity of
$$\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ (j^2\ \text{mod}\ p)>(k^2\ \text{mod}\ p)\right\}\right|$$
for any ...
- 15.6k
6
votes
1
answer
653
views
On permuted sum of squares of primes in a list
We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\...
- 13.9k
5
votes
2
answers
348
views
Permutations $\pi\in S_{p-1}$ with $\frac1{\pi(1)\pi(2)}+\frac1{\pi(2)\pi(3)}+\cdots+\frac1{\pi(p-2)\pi(p-1)}+\frac1{\pi(p-1)\pi(1)}\equiv0\pmod{p^2}$
A well known congruence of Wolstenholme states that
$$\frac1{1^2}+\frac1{2^2}+\cdots+\frac1{(p-1)^2}\equiv0\pmod{p}$$ for any prime $p>3$. For each $n=3,4,\ldots$ we clearly have
$$\frac1{1\times2}+...
- 15.6k
4
votes
1
answer
273
views
Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?
Note that for any permutation $\sigma\in S_5$ the product $\prod_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube.
Question. Let $n>5$ be an integer. Is the product $\prod_{k=1}^nk^{\sigma(k)}$ ...
- 15.6k
3
votes
2
answers
538
views
On the sum $\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$
Motivated by Question 316142 of mine, I consider the new sum
$$S(n):=\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$$
for any positive integer $n$, where $S_n$ is the symmetric group of all the ...
- 15.6k
3
votes
0
answers
131
views
Chen primes and permutations
In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...
- 15.6k
3
votes
0
answers
293
views
Primes arising from permutations (II)
In Question 315259 (cf. Primes arising from permutations) I asked a question on primes arising from permutations which looks quite challenging.
Here I pose a new question in this direction which does ...
- 15.6k
2
votes
1
answer
202
views
Endomorphism of the symmetric group of the set of positive integers via action on the prime numbers
For a positive integer $n$, let $p_n$ denote the $n$-th prime number.
Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$
be the monomorphism which maps a permutation $\sigma$ to ...
- 19.6k
2
votes
1
answer
106
views
Consecutive prime numbers in permutations of digits of the first consecutive positive integers
I have been toying for a while with the study of: in how many distinct primes and of which size can we divide permutations of digits of the first positive integers?
In this post I studied how many ...
- 189
2
votes
0
answers
192
views
A conjecture on crossing numbers related to primes
For a permutation $\sigma\in S_n$, its crossing number $\text{cr}(\sigma)$ is the number of pairs $\{i,j\}$ with $i,j\in\{1,\ldots,n\}$ such that
$$i<j\le\sigma(i)<\sigma(j)\ \ \text{or}\ \ \...
- 15.6k