A141189 Primes of the form x^2+7*x*y-y^2 (as well as of the form 7*x^2+9*x*y+y^2).

7, 11, 13, 17, 29, 37, 43, 47, 53, 59, 89, 97, 107, 113, 131, 149, 163, 197, 199, 211, 223, 227, 229, 241, 269, 271, 281, 293, 307, 311, 317, 331, 347, 367, 409, 431, 433, 439, 449, 461, 467, 487, 521, 523, 541, 547, 577, 587, 593, 599, 607, 619, 643, 647, 653

Offset

1,1

Comments

Discriminant = 53. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

A subsequence of (and may possibly coincide with) A038931. - R. J. Mathar, Jul 22 2008

References

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

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

Links

Table of n, a(n) for n=1..55.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Example

a(5) = 29 because we can write 29 = 3^2+7*3*1-1^2 (or 29 = 7*1^2+9*1*2+2^2).

Mathematica

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == x^2 + 7*x*y - y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Crossrefs

Cf. A038872 (d=5). A038873 (d=8). A038931, A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Primes in A243191.

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: A073704 A040108 A038931 * A191035 A038877 A019351

Adjacent sequences: A141186 A141187 A141188 * A141190 A141191 A141192

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