A141112 Primes of the form 2*x^2+5*x*y-5*y^2 (as well as of the form 7*x^2+11*x*y+2*y^2).

2, 5, 7, 13, 37, 47, 67, 73, 83, 97, 137, 163, 167, 193, 197, 223, 227, 293, 307, 317, 353, 383, 397, 457, 463, 487, 557, 577, 587, 593, 613, 617, 643, 683, 733, 743, 773, 787, 827, 853, 863, 877, 947, 967, 977, 983, 1033, 1087, 1097, 1103, 1123, 1163, 1217, 1237, 1307, 1367, 1373

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

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)

Example

a(4)=37 because we can write 37=2*6^2+5*6*7-5*7^2 (or 37=7*1^2+11*1*2+2*2^2)

Mathematica

Select[Prime[Range[250]], # == 5 || # == 13 || MatchQ[Mod[#, 65], Alternatives[2, 7, 8, 18, 28, 32, 33, 37, 47, 57, 58, 63]]&] (* Jean-François Alcover, Oct 28 2016 *)

Prog

(Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([2, 5, -5])
print(Q.represented_positives(1373, 'prime')) # Peter Luschny, Aug 12 2021

Crossrefs

Cf. A141111.

Sequence in context: A215210 A177997 A238776 * A053647 A023242 A164570

Adjacent sequences: A141109 A141110 A141111 * A141113 A141114 A141115

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