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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).
7
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
1,1
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
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
LINKS
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.
EXAMPLE
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.
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 *)
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.
Sequence in context: A060327 A202286 A285805 * A176371 A284531 A040987
KEYWORD
nonn
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
EXTENSIONS
Removed defective Mma program. - N. J. A. Sloane, Jun 06 2014
STATUS
approved