A141179 Primes of the form 3*x^2 + 2*x*y - 3*y^2 (as well as of the form 3*x^2 + 8*x*y + 2*y^2).

2, 3, 5, 13, 37, 43, 53, 67, 83, 107, 157, 163, 173, 197, 227, 277, 283, 293, 307, 317, 347, 373, 397, 443, 467, 523, 547, 557, 563, 587, 613, 643, 653, 677, 683, 733, 757, 773, 787, 797, 827, 853, 877, 883, 907, 947, 997, 1013, 1093, 1117, 1123, 1163, 1187, 1213, 1237, 1277

Offset

1,1

Comments

Discriminant = 40. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

For each term p > 5, p^2 == 13^2 (mod 240), and p is of the form 120*k +- b, where b = (13, 37, 43, 53). - Boyd Blundell, Jul 05 2021

References

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

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. OEIS wiki, June 2014.

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Example

13 is a term because we can write 13 = 3*2^2 + 2*2*1 - 3*1^2 (or 13 = 3*1^2 + 8*1*1 + 2*1^2).

Mathematica

Select[Prime[Range[250]], # == 2 || # == 5 || MatchQ[Mod[#, 40], Alternatives[3, 13, 27, 37]]&] (* Jean-François Alcover, Oct 28 2016 *)

Prog

(Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([3, 2, -3])
print(Q.represented_positives(1277, 'prime')) # Peter Luschny, Aug 12 2021

Crossrefs

Cf. A141180 (d=40). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Cf. also A243165.

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Sequence in context: A353582 A233515 A173268 * A215314 A079147 A339540

Adjacent sequences: A141176 A141177 A141178 * A141180 A141181 A141182

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 12 2008

Status

approved