|
%I #41 Feb 18 2022 03:53:37
%S 3,19,43,67,73,97,139,163,193,211,241,283,307,313,331,337,379,409,433,
%T 457,499,523,547,571,577,601,619,643,673,691,739,769,787,811,859,883,
%U 907,937,1009,1033,1051,1123,1129,1153,1171,1201,1249,1291,1297,1321,1459,1483,1489,1531
%N Primes of the form x^2+4*x*y-2*y^2 (as well as of the form 3*x^2+6*x*y+y^2).
%C Discriminant = 24. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
%C Also, primes of form u^2 - 6v^2. The transformation {u,v} = {x+2y,y} yields the form in the title. - _Tito Piezas III_, Dec 31 2008
%C Conjecture: this is also the list of primes that are simultaneously of the form x^2+2y^2 and of the form x^2+3y^2; that is, the intersection of A002476 and A033203. - _Zak Seidov_, Jun 07 2014
%C This is also the list of primes p such that p = 3 or p is congruent to 1 or 19 mod 24. - _Jean-François Alcover_, Oct 28 2016
%D Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H Juan Arias-de-Reyna, <a href="/A141170/b141170.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e a(2)=19 because we can write 19=3^2+4*3*1-2*1^2 (or 19=3*1^2+6*1*2+2^2)
%t xy[{x_, y_}]:={x^2 + 4 x y - 2 y^2, y^2 + 4 x y - 2 x^2}; Union[Select[Flatten[xy/@Subsets[Range[40], {2}]], #>0&&PrimeQ[#]&]] (* _Vincenzo Librandi_, Jun 09 2014 *)
%t Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 24], 1|19]&] (* _Jean-François Alcover_, Oct 28 2016 *)
%Y Cf. A141171 (d=24), A106950 (Primes of the form x^2+18y^2), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
%Y Cf. also A242661, A002476, A033203.
%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 (oscfalgan(AT)yahoo.es), Jun 12 2008
|
