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%I #34 Feb 17 2022 11:36:07
%S 37,67,107,137,139,151,233,269,293,317,349,367,491,601,691,823,839,
%T 863,877,881,929,941,971,1061,1069,1103,1163,1237,1259,1279,1283,1307,
%U 1373,1433,1489,1553,1601,1607,1627,1669,1693,1777,1783,1787,1847,1877,1973
%N Primes of the form x^2+12*x*y-y^2.
%C Discriminant = 148. Class = 3. 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 Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H Juan Arias-de-Reyna, <a href="/A141163/b141163.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="https://oeis.org/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 a(4)=137 because we can write 137= 3^2+12*3*4-4^2.
%t q := x^2 + 12*x*y - y^2; pmax = 2000; xmax = 100; ymin = -xmax; ymax = xmax; k = 1; prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, k*xmax}, {y, k *ymin, k *ymax}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; k++; Print["k = ", k, " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141163 = prms (* _Jean-François Alcover_, Oct 26 2016 *)
%o (Sage)
%o # The function binaryQF is defined in the link 'Binary Quadratic Forms'.
%o Q = binaryQF([1, 12, -1])
%o print(Q.represented_positives(1973, 'prime')) # _Peter Luschny_, Oct 26 2016
%Y Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141161 (d=148).
%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 (sergarmor(AT)yahoo.es), Jun 13 2008
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