A141176 Primes of the form 2*x^2 + 3*x*y - 3*y^2 (as well as of the form 6*x^2 + 9*x*y + 2*y^2).

2, 11, 17, 29, 41, 83, 101, 107, 131, 149, 167, 173, 197, 227, 233, 239, 263, 281, 293, 347, 359, 431, 461, 479, 491, 503, 557, 563, 569, 593, 659, 677, 701, 743, 761, 809, 821, 827, 857, 887, 941, 953, 1019, 1031, 1091, 1097, 1151, 1163, 1187, 1217, 1223, 1229, 1283, 1289, 1319, 1361

Offset: 1

Author

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

Keywords

• nonn

Comments

Discriminant = 33. 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.

These are primes = 11 or congruent to {2, 8, 17, 29, 32} mod 33. Note that the binary quadratic forms with discriminant 33 are in two classes as well as two genera, so there is one class in each genus. A141177 is in the other genus, with primes = 3 or congruent to {1, 4, 16, 25, 31} mod 33. - Jianing Song, Jul 30 2018

Examples

			a(3) = 17 because we can write 17 = 2*4^2 + 3*4*5 - 3*5^2 (or 17 = 6*1^2 + 9*1*1 + 2*1^2).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.
• D. B. Zagier, Zetafunktionen und quadratische Körper: Eine Einführung in die höhere Zahlentheorie, Springer-Verlag Berlin Heidelberg, 1981, DOI 10.1007/978-3-642-61829-1.

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)

Crossrefs

Cf. A141177 (d=33); A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13); A038889 (d=17); A141111, A141112 (d=65).

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[500]], # == 11 || MatchQ[Mod[#, 33], Alternatives[2, 8, 17, 29, 32]]&] (* Jean-François Alcover, Oct 28 2016 *)