A141187 Primes of the form -x^2+6*x*y+3*y^2 (as well as of the form 8*x^2+12*x*y+3*y^2).

3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Author

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

Keywords

• nonn

Comments

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]

Examples

			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).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

Extensions

More terms from Colin Barker, Apr 05 2015