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A141192 Primes of the form 3*x^2+3*x*y-4*y^2 (as well as of the form 8*x^2+11*x*y+2*y^2). 7


%S 2,3,29,41,53,59,71,89,107,113,167,173,179,227,257,269,281,293,317,

%T 383,401,431,449,509,521,563,569,599,641,659,677,683,743,773,797,827,

%U 839,857,863,887,911,941,953,971,977,983,1019,1091,1097,1181,1193,1229,1283,1307,1319

%N Primes of the form 3*x^2+3*x*y-4*y^2 (as well as of the form 8*x^2+11*x*y+2*y^2).

%C Discriminant = 57. 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

%C p = 3 and primes p = 2 mod 3 such that 57 is a square mod p. - Juan Arias-de-Reyna, Mar 20 2011.

%D Z. I. Borevich and I. R. Shafarevich, Number Theory.

%D D. B. Zagier, Zetafunktionen und quadratische Körper.

%H Juan Arias-de-Reyna, <a href="/A141192/b141192.txt">Table of n, a(n) for n = 1..10000</a>

%H 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)

%e a(6)=59 because we can write 59=3*7^2+3*7*8-4*8^2 (or 59=8*1^2+11*1*3+2*3^2)

%t Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 57], Alternatives[2, 8, 14, 29, 32, 41, 50, 53, 56]]&] (* _Jean-François Alcover_, Oct 28 2016 *)

%Y Cf. A141193 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

%Y Primes in A243192.

%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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Last modified July 9 14:56 EDT 2020. Contains 335543 sequences. (Running on oeis4.)