New answers tagged

0 votes

Prime numbers made of permutations of digits of consecutive positive integers

We knows that no prime of the form $1\_2\_...\_n$ has been found yet... at least for $n$ up to $10^6$ (see https://mathworld.wolfram.com/SmarandachePrime.html). Now, a partial answer to your question ...
user avatar
  • 1
1 vote

Prime numbers made of permutations of digits of consecutive positive integers

My impression is that the prime numbers in $S(3k+1)$ are exactly in the proportion they should be. Note that numbers in $S(3k+1)$ are smaller than $10^{D(3k+1)-1}$ because they have $D(3k+1)$ digits. ...
user avatar
3 votes

Vandermonde $V_n$ mod $n$

OP asked me to fill in the details of my comment, and in attempting to do so I realised that I claimed too much. However, a very similar argument proves a weaker result which is strong enough to ...
user avatar
  • 4,332
8 votes

Vandermonde $V_n$ mod $n$

Per comments above, for a counterexample we have with necessity $\pi_n=n$ and prime $n$. The case $n=2$ is trivial, so I assume that $n$ is an odd prime. The elements $U:=\{ 1,2,\dots,n-1\}$ form the ...
user avatar
5 votes
Accepted

Given $\pi$ permutation on $\{1,\dotsc,n\}$, what is the sign of a permutation of $\{2,\dotsc,\hat\jmath,\dotsc,n\}$?

$\DeclareMathOperator\sgn{sgn}$This is almost the same as your previous question, just with the order of the operations switched—whether you think of $\pi$ as ordering or disordering is just a matter ...
user avatar
  • 7,931
4 votes
Accepted

Sign of the permutation which brings a subsequence back to its original form

Imagine a bubble sort where you bring each element to its original position. $x_1$ would have taken $\pi^{-1}(1) - 1$ transpositions to bring it back to its original position. Let's perform those ...
user avatar
  • 7,931
5 votes

Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture

Fourier transform does it. Denote by $u_j$ $(j=0,1,\ldots,n-1$) the column-vector with coordinates $(x_{ji})_{1\leqslant i\leqslant n-1}$. Note that $u_0+u_1+\ldots+u_{n-1}=0$ and any $n-1$ vectors $...
user avatar
  • 88.6k
2 votes

Permuting subgroups with the same finite index

The main result of Lubotzky, Alexander; Mann, Avinoam; Segal, Dan, Finitely generated groups of polynomial subgroup growth, Isr. J. Math. 82, No. 1-3, 363-371 (1993). ZBL0811.20027. states that a ...
user avatar
0 votes

A determinant involving the cotangent function

The conjecture has been proved by my graduate student Han Wang and me via the Eigenvector-eigenvalue Identity, for our joint paper see http://arxiv.org/abs/2206.02589 .
user avatar

Top 50 recent answers are included