cp's OEIS Frontend

A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).

%I A141123 #37 Jun 22 2025 21:32:55
%S A141123 2,3,11,23,47,59,71,83,107,131,167,179,191,227,239,251,263,311,347,
%T A141123 359,383,419,431,443,467,479,491,503,563,587,599,647,659,683,719,743,
%U A141123 827,839,863,887,911,947,971,983,1019,1031,1091,1103,1151,1163,1187,1223
%N A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).
%C A141123 Discriminant = 12. 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 A141123 This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - _Jianing Song_, Jun 22 2025
%D A141123 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D A141123 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H A141123 Robert Israel, <a href="/A141123/b141123.txt">Table of n, a(n) for n = 1..6514</a>
%H A141123 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%e A141123 a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).
%p A141123 N:= 2000:
%p A141123 S:= NULL:
%p A141123 for xx from 1 to floor(2*sqrt(N/3)) do
%p A141123   for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do
%p A141123      S:= S, 3*xx^2-yy^2;
%p A141123 od od:
%p A141123 sort(convert(select(isprime,{S}),list)); # _Robert Israel_, Jul 20 2020
%t A141123 Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
%t A141123 (* or: *)
%t A141123 Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* _Jean-François Alcover_, Oct 25 2016, updated Oct 29 2016 *)
%Y A141123 Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
%Y A141123 Essentially the same as A068231 and A141187.
%Y A141123 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%Y A141123 Cf. A084917.
%K A141123 nonn
%O A141123 1,1
%A A141123 Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008
%E A141123 More terms from _Colin Barker_, Apr 05 2015