A141304 - OEIS

A141304

Primes of the form -2*x^2+6*x*y+3*y^2 (as well as of the form 7*x^2+12*x*y+3*y^2).

%I #22 Feb 17 2022 11:38:46

%S 3,7,43,67,103,127,163,223,283,307,367,463,487,523,547,607,643,727,

%T 787,823,883,907,967,1063,1087,1123,1303,1327,1423,1447,1483,1543,

%U 1567,1627,1663,1723,1747,1783,1867,1987,2083,2143,2203,2287,2347,2383,2467,2503,2647,2683

%N Primes of the form -2*x^2+6*x*y+3*y^2 (as well as of the form 7*x^2+12*x*y+3*y^2).

%C Discriminant = 60. Class = 4. 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 This is also the list of primes p such that p = 3 or p is congruent to 7 or 43 mod 60. - _Jean-François Alcover_, Oct 28 2016

%D Z. I. Borevich and I. R. Shafarevich, Number Theory.

%H Juan Arias-de-Reyna, <a href="/A141304/b141304.txt">Table of n, a(n) for n = 1..10000</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. 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(3)=43 because we can write 43=-2*1^2+6*1*3+3*3^2 (or 43=7*1^2+12*1*2+3*2^2).

%t Select[Prime[Range[500]], # == 3 || MatchQ[Mod[#, 60], 7|43]&] (* _Jean-François Alcover_, Oct 28 2016 *)

%Y Cf. A107152, A141302, A141303 (d=60).

%Y Primes in A243190.

%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 24 2008