A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).

2, 3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset

1,1

Comments

Discriminant = 12. 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.

References

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

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

Links

Robert Israel, Table of n, a(n) for n = 1..6514

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

Example

a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

Maple

N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
  for yy from ceil(sqrt(max(1, 3*xx^2-N))) to floor(sqrt(3)*xx) do
     S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime, {S}), list)); # Robert Israel, Jul 20 2020

Mathematica

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

Crossrefs

Cf. A068228 (d = 12), A068231 (Primes congruent to 11 (mod 12)), A141111, A141112 (d = 65).

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

Cf. A084917.

Sequence in context: A346605 A091310 A040994 * A263729 A246496 A119641

Adjacent sequences: A141120 A141121 A141122 * A141124 A141125 A141126

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 05 2008

Extensions

More terms from Colin Barker, Apr 05 2015

Status

approved