A270596 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.

131, 191, 239, 251, 311, 419, 431, 491, 599, 647, 659, 683, 743, 827, 911, 971, 1031, 1091, 1103, 1151, 1163, 1223, 1259, 1451, 1499, 1511, 1559, 1571, 1583, 1607, 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

Offset: 1

Author

Michel Marcus, Mar 20 2016

Keywords

• nonn

Comments

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.

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1983
• Peter Cameron's Blog, Permutation groups and regular semigroups, 2, Posted 22/08/2015.
• Michel Marcus, GAP program with issues

Crossrefs

Cf. A002476, A002144, A068231.

Programs

PARI

{ forprime(p=11, 10^6,
    if ( p%12 != 11, next() );
    for (c=2, p-2,
        my( v = vector(p-1) );
        my( g0 = Mod(c, p),  rc0 = znorder(g0) );
        if ( rc0 == p - 1,  next() );
        if ( znorder( -g0 ) == p - 1,  next() );
        my( g1 = Mod(c-1, p),  rc1 = znorder(g1) );
        if ( rc1 == p - 1,  next() );
        if ( znorder( -g1 ) == p - 1,  next() );
        if ( znorder( g0*g1 ) == p - 1,  next() );
        if ( znorder( -g0*g1 ) == p - 1,  next() );
        for (x0 = 0, rc0,
            my ( p0 = g0^x0,  z = p0 );
            for (x1 = 0, rc1,
                v[lift(z)] = 1;
                v[p - lift(z)] = 1;
                z * = g1;
            );
        );
        my( s = sum(k=1,#v,v[k]) );
        if ( s < p - 1,  print1(p,", "); break() );
    );
); } \\ Joerg Arndt, Mar 20 2016

Extensions

Terms > 500 by Joerg Arndt, Mar 20 2016