|
%I #17 Nov 15 2016 12:15:16
%S 131,191,239,251,311,419,431,491,599,647,659,683,743,827,911,971,1031,
%T 1091,1103,1151,1163,1223,1259,1451,1499,1511,1559,1571,1583,1607,
%U 1667,1787,1811,1847,1871,1931,2003,2087,2111,2243,2267,2339,2351,2399,2411,2423,2531,2591,2663,2687,2699,2711,2843,2927,2939,3011
%N Primes p congruent to 11 mod 12 (A068231), such that there exists a nonzero element c of GF(p), such that the element c, c-1 and -1 generate a proper subgroup of the multiplicative group.
%C P. Cameron shows that "primes congruent to 1 (mod 3) and greater than 7" (see A002476) and "primes congruent to 1 (mod 4) and greater than 5" (see A002144) also have this property.
%H Joerg Arndt, <a href="/A270596/b270596.txt">Table of n, a(n) for n = 1..1983</a>
%H Peter Cameron's Blog, <a href="https://cameroncounts.wordpress.com/2012/04/06/permutation-groups-and-regular-semigroups-2/">Permutation groups and regular semigroups, 2</a>, Posted 22/08/2015.
%H Michel Marcus, <a href="/A270596/a270596.g.txt">GAP program with issues</a>
%o (PARI)
%o { forprime(p=11, 10^6,
%o if ( p%12 != 11, next() );
%o for (c=2, p-2,
%o my( v = vector(p-1) );
%o my( g0 = Mod(c, p), rc0 = znorder(g0) );
%o if ( rc0 == p - 1, next() );
%o if ( znorder( -g0 ) == p - 1, next() );
%o my( g1 = Mod(c-1, p), rc1 = znorder(g1) );
%o if ( rc1 == p - 1, next() );
%o if ( znorder( -g1 ) == p - 1, next() );
%o if ( znorder( g0*g1 ) == p - 1, next() );
%o if ( znorder( -g0*g1 ) == p - 1, next() );
%o for (x0 = 0, rc0,
%o my ( p0 = g0^x0, z = p0 );
%o for (x1 = 0, rc1,
%o v[lift(z)] = 1;
%o v[p - lift(z)] = 1;
%o z * = g1;
%o );
%o );
%o my( s = sum(k=1,#v,v[k]) );
%o if ( s < p - 1, print1(p,", "); break() );
%o );
%o ); } \\ _Joerg Arndt_, Mar 20 2016
%Y Cf. A002476, A002144, A068231.
%K nonn
%O 1,1
%A _Michel Marcus_, Mar 20 2016
%E Terms > 500 by _Joerg Arndt_, Mar 20 2016
|
