A141193 Primes of the form -3*x^2+3*x*y+4*y^2 (as well as of the form 6*x^2+9*x*y+y^2).

7, 19, 43, 61, 73, 139, 157, 163, 199, 229, 271, 277, 283, 313, 349, 367, 397, 457, 463, 499, 541, 571, 577, 613, 619, 631, 643, 691, 709, 727, 733, 739, 757, 769, 823, 853, 859, 883, 919, 937, 967, 997, 1033, 1051, 1069, 1087, 1201, 1213, 1279, 1297, 1303, 1327, 1423, 1429

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008

Keywords

• nonn

Comments

Discriminant = 57. 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

p = 19 and primes p = 1 mod 3 such that 57 is a square mod p. - Juan Arias-de-Reyna, Mar 20 2011

Examples

			a(2)=19 because we can write 19=-3*1^2+3*1*2+4*2^2
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A141192 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Primes in A243193.

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

Select[Prime[Range[250]], # == 19 || MatchQ[Mod[#, 57], Alternatives[1, 4, 7, 16, 25, 28, 43, 49, 55]]&] (* Jean-François Alcover, Oct 28 2016 *)