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

1,1

Comments

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

References

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

Links

Table of n, a(n) for n=1..50.

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

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).

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 *)

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).

Sequence in context: A149661 A146130 A026619 * A273422 A192146 A273607

Adjacent sequences: A142953 A142954 A142955 * A142957 A142958 A142959

Keyword

nonn

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

Extensions

More terms from Colin Barker, Apr 05 2015

Edited by M. F. Hasler, Feb 18 2022

Status

approved