cp's OEIS Frontend

A141111 Primes of the form 4*x^2+x*y-4*y^2 (as well as of the form 4*x^2+9*x*y+y^2).

%I A141111 #46 Feb 28 2020 08:18:55
%S A141111 29,61,79,101,131,139,179,181,191,199,211,251,269,311,389,419,439,491,
%T A141111 521,569,571,599,601,641,659,701,719,751,809,829,859,881,911,919,971,
%U A141111 991,1031,1039,1049,1069,1091,1109,1171,1231,1249,1291,1301,1361,1381,1429,1439,1459,1481,1499,1511,1531
%N A141111 Primes of the form 4*x^2+x*y-4*y^2 (as well as of the form 4*x^2+9*x*y+y^2).
%C A141111 Both have discriminant = 65. 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.
%D A141111 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D A141111 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H A141111 Juan Arias-de-Reyna, <a href="/A141111/b141111.txt">Table of n, a(n) for n = 1..10000</a>
%H A141111 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms#Implementation">Binary Quadratic Forms</a>
%H A141111 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 A141111 a(3)=79 because we can write 79=4*5^2+5*3-4*3^2 (or 79=4*2^2+9*2*3+3^2).
%t A141111 Select[Prime[Range[250]], MatchQ[Mod[#, 65], Alternatives[1, 4, 9, 14, 16, 29, 36, 49, 51, 56, 61, 64]]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%o A141111 (Sage) # uses[binaryQF]
%o A141111 # The function binaryQF is defined in the link 'Binary Quadratic Forms'.
%o A141111 Q = binaryQF([4, 1, -4])
%o A141111 print(Q.represented_positives(1531, 'prime')) # _Peter Luschny_, Oct 27 2016
%Y A141111 Cf. A141112, A243170.
%K A141111 nonn
%O A141111 1,1
%A A141111 Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 04 2008, Jun 05 2008