A141163 Primes of the form x^2+12*x*y-y^2.

37, 67, 107, 137, 139, 151, 233, 269, 293, 317, 349, 367, 491, 601, 691, 823, 839, 863, 877, 881, 929, 941, 971, 1061, 1069, 1103, 1163, 1237, 1259, 1279, 1283, 1307, 1373, 1433, 1489, 1553, 1601, 1607, 1627, 1669, 1693, 1777, 1783, 1787, 1847, 1877, 1973

Offset: 1

Author

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

Keywords

• nonn

Comments

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.

Examples

			a(4)=137 because we can write 137= 3^2+12*3*4-4^2.
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• Peter Luschny, Binary Quadratic Forms
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141161 (d=148).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Programs

Mathematica

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 *)

Sage

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 12, -1])
print(Q.represented_positives(1973, 'prime')) # Peter Luschny, Oct 26 2016