A141190 Primes of the form 2*x^2+4*x*y-5*y^2 (as well as of the form 2*x^2+8*x*y+y^2).

2, 11, 43, 67, 107, 113, 137, 163, 179, 193, 211, 233, 281, 331, 337, 347, 379, 401, 443, 449, 457, 491, 499, 547, 569, 571, 617, 641, 659, 673, 683, 739, 809, 827, 883, 907, 947, 953, 977, 1009, 1019, 1033, 1051, 1129, 1163, 1171, 1187, 1201, 1283, 1289

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), 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.

Also primes of the form x^2 + 6xy - 5y^2, cf. A243186. - N. J. A. Sloane, Jun 05 2014

Examples

			a(3) = 43 is in the sequence because we can write 43 = 2*4^2 + 4*4*1 - 5*1^2, or 43 = 2*3^2 + 8*3*1 + 1^2.
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• 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.

Crossrefs

Cf. A141191 (d=56) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Cf. A243186.

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

xy[{x_, y_}]:={2 x^2 + 4 x y - 5 y^2, 2 y^2 + 4 x y - 5 x^2}; Union[Select[Flatten[xy/@Subsets[Range[50], {2}]], #>0&&PrimeQ[#]&]] (* Vincenzo Librandi, Jun 09 2014 *)