cp's OEIS Frontend

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).

%I A141180 #45 Feb 18 2022 15:30:58
%S A141180 31,41,71,79,89,151,191,199,239,241,271,281,311,359,401,409,431,439,
%T A141180 449,479,521,569,599,601,631,641,719,751,761,769,809,839,881,911,919,
%U A141180 929,991,1009,1031,1039,1049,1129,1151,1201,1231,1249,1279,1289,1319,1321,1361,1399,1409,1439
%N 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).
%C A141180 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.
%C A141180 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
%C A141180 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
%C A141180 All primes p such that (p^2 - 1)/24 mod 10 = 0. See A024702.  - _Richard R. Forberg_, Aug 27 2013
%D A141180 Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H A141180 Juan Arias-de-Reyna, <a href="/A141180/b141180.txt">Table of n, a(n) for n = 1..10000</a>
%H A141180 N. J. A. Sloane et al., <a href="/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a>: Index to related sequences, programs, references. OEIS wiki, June 2014.
%H A141180 D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e A141180 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.
%t A141180 Take[Select[Union[Flatten[Table[Abs[a^2 - 10b^2], {a, 0, 49}, {b, 0, 49}]]], PrimeQ], 50] (* _Alonso del Arte_, Jul 22 2012 *)
%t A141180 Select[Prime[Range[250]], MatchQ[Mod[#, 40], Alternatives[1, 9, 31, 39]]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%Y A141180 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.
%Y A141180 Cf. also A242664.
%Y A141180 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K A141180 nonn
%O A141180 1,1
%A A141180 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
%E A141180 Removed defective Mma program. - _N. J. A. Sloane_, Jun 06 2014