cp's OEIS Frontend

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

%I A106865 #45 Jun 22 2025 17:34:33
%S A106865 2,3,7,23,43,47,67,83,103,107,127,163,167,223,227,263,283,307,347,367,
%T A106865 383,443,463,467,487,503,523,547,563,587,607,643,647,683,727,743,787,
%U A106865 823,827,863,883,887,907,947,967,983,1063,1087,1103,1123,1163,1187
%N A106865 Primes of the form 2x^2 + 2xy + 3y^2.
%C A106865 Discriminant = -20.
%C A106865 Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
%C A106865 Primes congruent to 2, 3, 7 modulo 20. - _Michael Somos_, Aug 13 2006
%C A106865 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
%D A106865 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
%H A106865 Vincenzo Librandi and Ray Chandler, <a href="/A106865/b106865.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi]
%H A106865 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F A106865 Complement(A000040, A020669).
%e A106865 x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
%e A106865 x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.
%p A106865 select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # _Robert Israel_, Dec 23 2015
%t A106865 QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)
%o A106865 (PARI) is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ _Charles R Greathouse IV_, Feb 09 2017
%Y A106865 For n > 1, a(n) = A122870(n-1). Cf. A122870, A106864.
%K A106865 nonn,easy
%O A106865 1,1
%A A106865 _T. D. Noe_, May 09 2005