All Questions
8 questions
0
votes
1
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166
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A property related to permutations with coprime adjacent values
Sequence A76220 of OEIS enumerates (up to $n=25$) the number $a_n$
of permutations $\sigma$ of $\lbrace 1,\ldots,n\rbrace$ such that
$\sigma(i)$ and $\sigma(i+1)$ are coprime for $i=1,\ldots,n-1$.
All ...
- 17.9k
10
votes
1
answer
694
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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 $...
- 4,928
2
votes
0
answers
192
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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
3
votes
0
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131
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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
19
votes
1
answer
3k
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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.
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- 15.6k
3
votes
0
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293
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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
7
votes
1
answer
531
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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
37
votes
2
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3k
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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 ...
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