cp's OEIS Frontend

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

%I A316565 #22 Dec 19 2019 22:08:28
%S A316565 1,3,8,6,24,24,48,12,24,60,120,24,168,48,60,24,288,24,360,60,168,330,
%T A316565 528,24,120,168,72,84,840,120,960,48,440,816,420,36,1368,360,312,60,
%U A316565 1680,168,1848,330,180,1518,2208,48,336,300,816,168,2808,72,1320,168
%N A316565 Maximum order of an element of the general linear group GL(2, Z(n)).
%F A316565 Conjecture: a(p) = (p-1)*(p+1) for prime p.
%F A316565 From _Robert Israel_, Dec 19 2019: (Start)
%F A316565 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.
%F A316565 a(p^k) <= (p^2-1) p^(k-1).
%F A316565 If m and n are coprime, a(m*n) <= a(m)*a(n). (End)
%o A316565 (GAP) Concatenation([1], List([2..10], n->Maximum(List(GL(2, Integers mod n), Order))));
%o A316565 (PARI)
%o A316565 MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
%o A316565 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}
%Y A316565 Row lengths of A316566.
%Y A316565 Cf. A000252, A316563.
%K A316565 nonn
%O A316565 1,2
%A A316565 _Andrew Howroyd_, Jul 06 2018