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A328643 Positive integers m such that the matrix E_m has order 3^m-1 in GL_m(3) where E_m is the m X m invertible tridiagonal matrix with all nonzero entries equal to 1 except for the (m,m) entry that is equal to 2. 1
1, 3, 5, 9, 11, 23, 29, 35, 39, 41, 53, 65, 69, 81, 83, 89, 95, 99, 105, 113, 119, 131, 155, 173, 179, 189, 191, 209, 221, 231, 233, 239, 243, 251, 281, 293, 299, 303, 323, 329, 359, 371, 375, 411, 413, 419, 429, 431, 443, 453, 491 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The cyclic subgroups of GL_m(q) of order q^m-1 are called Singer cycles.

LINKS

Table of n, a(n) for n=1..51.

M. Farrokhi D. G., Lattice paths inside a table: Rows and columns linear combinations, arXiv:1910.09844 [math.CO], 2019.

W. M. Kantor, Linear groups containing a Singer cycle, J. Algebra 62(1) (1980), 232-234.

EXAMPLE

For n = 3 the a(3) = 5 solution is the matrix E_5 =

[ [ 1 1 0 0 0 ],

  [ 1 1 1 0 0 ],

  [ 0 1 1 1 0 ],

  [ 0 0 1 1 1 ],

  [ 0 0 0 1 2 ] ]

since the matrix E_5 has order 3^5 - 1 = 242 in GL_5(3).

PROG

(GAP)

EMatrix := function(n, q)

local M, i;

M := NullMat(n, n, GF(q));

for i in [2..n] do

  M[i - 1][i - 1] := Z(q) ^ 0;

  M[i - 1][i] := Z(q) ^ 0;

  M[i][i - 1] := Z(q) ^ 0;

od;

M[n][n] := 2 * Z(q) ^ 0;

return M;

end;

for n in [1..100] do

  M := EMatrix(n, 3);

  if Determinant(M) <> 0 * Z(3) and Order(M) = 3 ^ n - 1 then

    Print(n, "\n");

  fi;

od;

(PARI)

E(m)={matrix(m, m, i, j, (i==m&&j==m) + (abs(i-j)<=1))}

is(m, b)={my(ID=matid(m), M=Mod(E(m), b), e=b^m-1); if(M^e==ID, fordiv(e, d, if(d<e && M^d==ID, return(0))); 1, 0)}

for(m=1, 100, if(m<>2&&is(m, 3), print1(m, ", "))) \\ Andrew Howroyd, Dec 21 2019

CROSSREFS

Cf. A328642.

Sequence in context: A092917 A256220 A163778 * A160358 A319084 A120806

Adjacent sequences:  A328640 A328641 A328642 * A328644 A328645 A328646

KEYWORD

nonn

AUTHOR

M. Farrokhi D. G., Oct 23 2019

STATUS

approved

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Last modified September 18 22:48 EDT 2020. Contains 337174 sequences. (Running on oeis4.)