A141191 Primes of the form -2*x^2+4*x*y+5*y^2 (as well as of the form 10*x^2+16*x*y+5*y^2).

5, 7, 13, 31, 47, 61, 101, 103, 157, 167, 173, 181, 199, 223, 229, 269, 271, 293, 311, 349, 367, 383, 397, 439, 461, 479, 503, 509, 607, 647, 661, 677, 719, 727, 733, 773, 797, 829, 839, 853, 887, 941, 983, 997, 1013, 1021, 1039, 1063, 1069, 1109, 1151, 1181

Offset: 1

Author

Laura Caballero Fernandez,Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

Keywords

• nonn

Comments

Discriminant = 56. 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.

Also primes of the form -x^2+6xy+5y^2. cf. A243187.

Examples

			a(4)=31 because we can write 31=-2*7^2+4*7*3+5*3^2 (or 31=10*1^2+16*1*1+5*1^2).
		

References

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

Links

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

Crossrefs

Cf. A141190 (d=56) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Cf. A243187, A243186, A141190.

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

Programs

Mathematica

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -2*x^2 + 4*x*y + 5*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Extensions

More terms from Colin Barker, Apr 05 2015