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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 April 23 22:36 EDT 2024. Contains 371917 sequences. (Running on oeis4.)