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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)