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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). 43

%I

%S 2,3,11,23,47,59,71,83,107,131,167,179,191,227,239,251,263,311,347,

%T 359,383,419,431,443,467,479,491,503,563,587,599,647,659,683,719,743,

%U 827,839,863,887,911,947,971,983,1019,1031,1091,1103,1151,1163,1187,1223

%N 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 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.

%D Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

%D D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

%H Robert Israel, <a href="/A141123/b141123.txt">Table of n, a(n) for n = 1..6514</a>

%H 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 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 N:= 2000:

%p S:= NULL:

%p for xx from 1 to floor(2*sqrt(N/3)) do

%p for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do

%p S:= S, 3*xx^2-yy^2;

%p od od:

%p sort(convert(select(isprime,{S}),list)); # _Robert Israel_, Jul 20 2020

%t 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 (* or: *)

%t Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* _Jean-François Alcover_, Oct 25 2016, updated Oct 29 2016 *)

%Y Cf. A068228 (d = 12), A068231 (Primes congruent to 11 (mod 12)), A141111, A141112 (d = 65).

%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

%Y Cf. A084917.

%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 05 2008

%E More terms from _Colin Barker_, Apr 05 2015

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)