A141111 Primes of the form 4*x^2+x*y-4*y^2 (as well as of the form 4*x^2+9*x*y+y^2).

29, 61, 79, 101, 131, 139, 179, 181, 191, 199, 211, 251, 269, 311, 389, 419, 439, 491, 521, 569, 571, 599, 601, 641, 659, 701, 719, 751, 809, 829, 859, 881, 911, 919, 971, 991, 1031, 1039, 1049, 1069, 1091, 1109, 1171, 1231, 1249, 1291, 1301, 1361, 1381, 1429, 1439, 1459, 1481, 1499, 1511, 1531

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 04 2008, Jun 05 2008

Keywords

• nonn

Comments

Both have discriminant = 65. 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.

Examples

			a(3)=79 because we can write 79=4*5^2+5*3-4*3^2 (or 79=4*2^2+9*2*3+3^2).
		

References

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

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• Peter Luschny, Binary Quadratic Forms
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A141112, A243170.

Programs

Mathematica

Select[Prime[Range[250]], MatchQ[Mod[#, 65], Alternatives[1, 4, 9, 14, 16, 29, 36, 49, 51, 56, 61, 64]]&] (* Jean-François Alcover, Oct 28 2016 *)

Sage

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([4, 1, -4])
print(Q.represented_positives(1531, 'prime')) # Peter Luschny, Oct 27 2016