A142956 Primes of the form -3*x^2 + 4*x*y + 5*y^2 (as well as of the form 6*x^2 + 10*x*y + y^2).

5, 17, 61, 73, 101, 137, 149, 157, 197, 229, 233, 277, 313, 349, 353, 389, 397, 457, 461, 541, 557, 577, 593, 613, 617, 653, 701, 709, 733, 757, 761, 769, 809, 821, 853, 881, 929, 937, 997, 1013, 1033, 1049, 1061, 1069, 1109, 1201, 1213, 1217, 1277, 1289

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (laucabfer(AT)alum.us.es), Jul 14 2008

Keywords

• nonn

Comments

Discriminant = 76. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

Examples

			a(2) = 17 because we can write 17 = -3*3^2 + 4*3*2 + 5*2^2 (or 17 = 6*1^2 + 10*1*1 + 1^2).
		

References

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

Links

• 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. A142955 (d=76). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

Programs

Mathematica

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -3*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

Edited by M. F. Hasler, Feb 18 2022