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

1,1

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

Prog

(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

Crossrefs

Cf. A002476, A002144, A068231.

Sequence in context: A050261 A180544 A032750 * A085414 A068686 A082807

Adjacent sequences: A270593 A270594 A270595 * A270597 A270598 A270599

Keyword

nonn

Author

Michel Marcus, Mar 20 2016

Extensions

Terms > 500 by Joerg Arndt, Mar 20 2016

Status

approved