1,1

Discriminant = 48. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

Values of the quadratic form are {0,3,8,11} mod 12, so all values with the exception of 3 are also in A068231. - R. J. Mathar, Jul 30 2008

Is this the same sequence (apart from the initial 3) as A068231? [Yes, since the orders of imaginary quadratic fields with discriminant 48 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025]

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

a(3)=23 because we can write 23= -1^2+6*1*2+3*2^2 (or 23=8*1^2+12*1*1+3*1^2).

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).

Essentially the same as A068231 and A141123.

Cf. A243169.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Sequence in context: A078723 A294368 A296556 * A107138 A145473 A335677

Adjacent sequences: A141184 A141185 A141186 * A141188 A141189 A141190

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

More terms from Colin Barker, Apr 05 2015

approved