A141180 Primes of the form x^2+6*x*y-y^2 (as well as of the form 6*x^2+8*x*y+y^2).

31, 41, 71, 79, 89, 151, 191, 199, 239, 241, 271, 281, 311, 359, 401, 409, 431, 439, 449, 479, 521, 569, 599, 601, 631, 641, 719, 751, 761, 769, 809, 839, 881, 911, 919, 929, 991, 1009, 1031, 1039, 1049, 1129, 1151, 1201, 1231, 1249, 1279, 1289, 1319, 1321, 1361, 1399, 1409, 1439

Offset: 1

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

Keywords

• nonn

Comments

Discriminant = 40. 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.

Also primes of form 10*u^2 - v^2. The transformation {u, v} = {-x, 3*x-y} yields the form in the title, and primes of form U^2 - 10*V^2, with transformation {U, V} = {x+3*y, y}. - Juan Arias-de-Reyna, Mar 19 2011

Therefore, these primes are composite in Q(sqrt(10)), as they can be factored thus: (-u + v*sqrt(10))*(u + v*sqrt(10)). - Alonso del Arte, Jul 22 2012

All primes p such that (p^2 - 1)/24 mod 10 = 0. See A024702. - Richard R. Forberg, Aug 27 2013

Examples

			a(2) = 41 because we can write 41 = 3^2 + 6*3*2 - 2^2 (or 41 = 6*2^2 + 8*2*1 + 1^2). Furthermore, notice that (-7 + 3*sqrt(10))(7 + 3*sqrt(10)) = 41.
		

References

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

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
• 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. A141179 (d=40) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65). A024702.

Cf. also A242664.

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

Take[Select[Union[Flatten[Table[Abs[a^2 - 10b^2], {a, 0, 49}, {b, 0, 49}]]], PrimeQ], 50] (* Alonso del Arte, Jul 22 2012 *)
Select[Prime[Range[250]], MatchQ[Mod[#, 40], Alternatives[1, 9, 31, 39]]&] (* Jean-François Alcover, Oct 28 2016 *)

Extensions

Removed defective Mma program. - N. J. A. Sloane, Jun 06 2014