A141181 - OEIS

%I #28 Jun 29 2024 11:07:27

%S 2,5,23,31,37,41,43,59,61,73,83,103,107,113,127,131,139,163,173,197,

%T 223,241,251,269,271,277,283,307,337,349,353,359,367,373,379,389,401,

%U 409,419,431,433,443,449,461,467,487,491,523,541,569,599,607,613,617,619

%N Primes of the form 2*x^2+3*x*y-4*y^2 (as well as of the form 2*x^2+7*x*y+y^2).

%C Discriminant = 41. Class = 1. 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 It appears that this is the same as "Primes that are squares (mod 41)", cf. A038919 and A373751. - _M. F. Hasler_, Jun 29 2024

%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 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) = 23 because we can write 23 = 2*3^2+3*3*1-4*1^2 (or 23 = 2*2^2+7*2*1+1^2).

%t Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 2*x^2 + 3*x*y - 4*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Oct 25 2016 *)

%o (PARI) select(p->isprime(p)&&qfbsolve(Qfb(1,7,2),p),[1..1500]) \\ This is to provide a generic characteristic function ("is_A141181") as 1st arg of select(), there are other ways to produce the sequence more efficiently. - _M. F. Hasler_, Jan 15 2016

%Y Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).

%Y Primes in A035269.

%Y A subsequence of (and may possibly coincide with) A038919. - _R. J. Mathar_, Jul 22 2008

%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