All Questions
14 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
3
votes
0
answers
193
views
A conjectural lower bound for $|\{\sum_{k=1}^nka_k:\ a_1,\ldots,a_n\ \text{are distinct elements of }\ A\}|$
Motivated by Question 315568 of mine, I'm interested in the set
$$S(n):=\bigg\{\sum_{k=1}^n k\pi(k):\ \pi\in S_n\bigg\}.$$
It is easy to see that
$$S(1)=\{1\},\ S(2)=\{4,5\}\ \text{and}\ S(3)=\{10,...
- 15.6k
7
votes
1
answer
531
views
Primes arising from permutations
Recently, Paul Bradley proved in arXiv:1809.01012 that for any positive integer $n$ there is a permutation $\pi_n$ of $\{1,\ldots,n\}$ such that $k+\pi_n(k)$ is prime for every $k=1,\ldots,n$ (cf. ...
- 15.6k
21
votes
1
answer
1k
views
Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$
Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Motivated by Question 315568 (http://mathoverflow.net/questions/315568), here I pose the following question.
QUESTION: ...
- 15.6k
19
votes
1
answer
3k
views
A mysterious connection between primes and squares
Motivated by two previous questions of mine (cf. Primes arising from permutations and Primes arising from permutations (II)), here I ask a curious question which connects primes with squares.
...
- 15.6k
12
votes
0
answers
643
views
Wilf's conjecture: complementary Bell numbers
The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by
$$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$
Definition. Fix an integer $m\geq0$....
- 43.2k
10
votes
1
answer
694
views
Prime numbers from permutation
Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
- 4,938
6
votes
2
answers
395
views
On the parity of $|\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ \ j(j+1)\ \text{mod}\ p\,>\,k(k+1)\ \text{mod}\ p\}|$ with $p$ prime
Let $p=2n+1$ be an odd prime, and let $a_1<\ldots<a_{n}$ be all the quadratic residues mod $p$ among $1,\ldots,p-1$. For $a\in\mathbb Z$ let $\{a\}_p$ be the least nonnegative residue of $a$ ...
- 15.6k
3
votes
0
answers
151
views
Combinatorial interpretation of inverse modulo $2$ binomial transform of A284005
My question is related to the following:
Sum with products turned into subsequences
We have an identity
$$a(n, -1) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}(-1)^{\operatorname{wt}(n)-\...
- 4,938
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
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
2
votes
1
answer
172
views
Permutation and its binary analog
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,938
2
votes
1
answer
364
views
Odd permutations $\tau\in S_n$ with $\sum_{k=1}^nk\tau(k)$ an odd square
For any positive integer $n$, as usual we let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
QUESTION: Is it true that for each integer $n>3$ there is an odd permutation ...
- 15.6k
-1
votes
1
answer
395
views
Odd & even permutations and unit fractions
One more motivated by recent questions of Zhi-Wei Sun.
Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.
Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
- 4,589