A316565 Maximum order of an element of the general linear group GL(2, Z(n)).

1, 3, 8, 6, 24, 24, 48, 12, 24, 60, 120, 24, 168, 48, 60, 24, 288, 24, 360, 60, 168, 330, 528, 24, 120, 168, 72, 84, 840, 120, 960, 48, 440, 816, 420, 36, 1368, 360, 312, 60, 1680, 168, 1848, 330, 180, 1518, 2208, 48, 336, 300, 816, 168, 2808, 72, 1320, 168

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Formula

Conjecture: a(p) = (p-1)*(p+1) for prime p.

From Robert Israel, Dec 19 2019: (Start)

The conjecture is true. In fact for T in GL(2,Z(p)), the order of T divides p*(p-1) if the characteristic polynomial of T splits over Z(p) and p^2-1 if it doesn't; moreover, if T is the companion matrix of the minimal polynomial of a primitive element of GF(p^2), the order is p^2-1.

a(p^k) <= (p^2-1) p^(k-1).

If m and n are coprime, a(m*n) <= a(m)*a(n). (End)

Prog

(GAP) Concatenation([1], List([2..10], n->Maximum(List(GL(2, Integers mod n), Order))));
(PARI)
MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}
a(n)={my(m=0); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(gcd(lift(matdet(M)), n)==1, m=max(m, MatOrder(M))))))); m}

Crossrefs

Row lengths of A316566.

Cf. A000252, A316563.

Sequence in context: A093725 A347118 A306933 * A345052 A011413 A296469

Adjacent sequences: A316562 A316563 A316564 * A316566 A316567 A316568

Keyword

nonn

Author

Andrew Howroyd, Jul 06 2018

Status

approved