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URL: https://oeis.org/A141111

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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).
45
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
1,1
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.
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
LINKS
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
EXAMPLE
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).
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 *)
PROG
(SageMath) # 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
CROSSREFS
Sequence in context: A042680 A286005 A132770 * A122114 A173032 A142047
KEYWORD
nonn
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
STATUS
approved