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%I #38 Feb 18 2022 15:30:50
%S 2,3,5,13,37,43,53,67,83,107,157,163,173,197,227,277,283,293,307,317,
%T 347,373,397,443,467,523,547,557,563,587,613,643,653,677,683,733,757,
%U 773,787,797,827,853,877,883,907,947,997,1013,1093,1117,1123,1163,1187,1213,1237,1277
%N Primes of the form 3*x^2 + 2*x*y - 3*y^2 (as well as of the form 3*x^2 + 8*x*y + 2*y^2).
%C Discriminant = 40. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
%C For each term p > 5, p^2 == 13^2 (mod 240), and p is of the form 120*k +- b, where b = (13, 37, 43, 53). - _Boyd Blundell_, Jul 05 2021
%D Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H Juan Arias-de-Reyna, <a href="/A141179/b141179.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms#Implementation">Binary Quadratic Forms</a>
%H N. J. A. Sloane et al., <a href="/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a>: Index to related sequences, programs, references. OEIS wiki, June 2014.
%H D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e 13 is a term because we can write 13 = 3*2^2 + 2*2*1 - 3*1^2 (or 13 = 3*1^2 + 8*1*1 + 2*1^2).
%t Select[Prime[Range[250]], # == 2 || # == 5 || MatchQ[Mod[#, 40], Alternatives[3, 13, 27, 37]]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%o (Sage) # uses[binaryQF]
%o # The function binaryQF is defined in the link 'Binary Quadratic Forms'.
%o Q = binaryQF([3, 2, -3])
%o print(Q.represented_positives(1277, 'prime')) # _Peter Luschny_, Aug 12 2021
%Y Cf. A141180 (d=40). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
%Y Cf. also A243165.
%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K nonn
%O 1,1
%A Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina, and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008
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