cp's OEIS Frontend

A141112 Primes of the form 2*x^2+5*x*y-5*y^2 (as well as of the form 7*x^2+11*x*y+2*y^2).

%I A141112 #27 Dec 08 2024 11:06:17
%S A141112 2,5,7,13,37,47,67,73,83,97,137,163,167,193,197,223,227,293,307,317,
%T A141112 353,383,397,457,463,487,557,577,587,593,613,617,643,683,733,743,773,
%U A141112 787,827,853,863,877,947,967,977,983,1033,1087,1097,1103,1123,1163,1217,1237,1307,1367,1373
%N A141112 Primes of the form 2*x^2+5*x*y-5*y^2 (as well as of the form 7*x^2+11*x*y+2*y^2).
%C A141112 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 A141112 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D A141112 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H A141112 Juan Arias-de-Reyna, <a href="/A141112/b141112.txt">Table of n, a(n) for n = 1..10000</a>
%H A141112 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms#Implementation">Binary Quadratic Forms</a>
%H A141112 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 A141112 a(4)=37 because we can write 37=2*6^2+5*6*7-5*7^2 (or 37=7*1^2+11*1*2+2*2^2)
%t A141112 Select[Prime[Range[250]], # == 5 || # == 13 || MatchQ[Mod[#, 65], Alternatives[2, 7, 8, 18, 28, 32, 33, 37, 47, 57, 58, 63]]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%o A141112 (Sage) # uses[binaryQF]
%o A141112 # The function binaryQF is defined in the link 'Binary Quadratic Forms'.
%o A141112 Q = binaryQF([2, 5, -5])
%o A141112 print(Q.represented_positives(1373, 'prime')) # _Peter Luschny_, Aug 12 2021
%Y A141112 Cf. A141111.
%Y A141112 Subsequence of A243171.
%K A141112 nonn
%O A141112 1,1
%A A141112 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