A106865 Primes of the form 2x^2 + 2xy + 3y^2.

2, 3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187

Offset: 1

Author

T. D. Noe, May 09 2005

Keywords

• nonn
• easy

Comments

Discriminant = -20.

Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.

Primes congruent to 2, 3, 7 modulo 20. - Michael Somos, Aug 13 2006

In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - Alonso del Arte, Dec 19 2015

Examples

			x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.
		

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

For n > 1, a(n) = A122870(n-1). Cf. A122870, A106864.

Programs

Maple

select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # Robert Israel, Dec 23 2015

Mathematica

QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)

PARI

is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ Charles R Greathouse IV, Feb 09 2017

Formula

Complement(A000040, A020669).