A141302 Primes of the form -x^2+6*x*y+6*y^2 (as well as of the form 11*x^2+18*x*y+6*y^2).

11, 59, 71, 131, 179, 191, 239, 251, 311, 359, 419, 431, 479, 491, 599, 659, 719, 839, 911, 971, 1019, 1031, 1091, 1151, 1259, 1319, 1439, 1451, 1499, 1511, 1559, 1571, 1619, 1811, 1871, 1931, 1979, 2039, 2099, 2111, 2339, 2351, 2399, 2411, 2459, 2531, 2579, 2591, 2699, 2711

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008

Keywords

• nonn

Comments

Discriminant = 60. Class number = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1 if they are primitive.

The Pell form X^2 - 15*Y^2 represents the negative primes -a(n), for n >= 1. - Wolfdieter Lang, Nov 28 2024

Examples

			a(3)=71 because we can write 71=-1^2+6*1*3+6*3^2 (or 71=11*1^2+18*1*2+6*2^2).
		

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. A107152, A141303, A141304 (d=60).

Primes in A237606.

Programs

Mathematica

Reap[For[p = 2, p < 3000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 6*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Formula

Primes congruent to {11, 59} (mod 60). -Wolfdieter Lang, Dec 22 2024

Extensions

Offset corrected by Mohammed Yaseen, May 20 2023