cp's OEIS Frontend

A141215 Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2).

%I A141215 #40 Feb 18 2022 15:31:31
%S A141215 3,5,13,19,41,47,61,73,83,97,103,107,109,113,127,131,137,149,163,167,
%T A141215 179,197,199,229,239,241,257,263,269,271,283,293,317,347,353,367,379,
%U A141215 431,439,443,449,461,463,479,487,491,503,563,569,571,601,607,613,619
%N A141215 Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2).
%C A141215 Discriminant = 61. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
%C A141215 A subsequence of (and may possibly coincide with) A038941. - _R. J. Mathar_, Jul 22 2008
%C A141215 3*x^2+5*x*y-3*y^2 and 5*x^2+9*x*y+y^2 are equivalent forms.
%C A141215 Also, primes of the form x^2 - 61y^2, of discriminant 244.
%D A141215 Z. I. Borevich and I. R. Shafarevich, Number Theory.
%H A141215 Robert Israel, <a href="/A141215/b141215.txt">Table of n, a(n) for n = 1..10000</a>
%H A141215 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 A141215 D. B. Zagier, <a href="https://doi.org/10.1007/978-3-642-61829-1">Zetafunktionen und quadratische Körper</a>, Springer, 1981.
%e A141215 a(8) = 73 because we can write 73 = 3*4^2+5*4*5-3*5^2 (or 73 = 5*3^2+9*3*1+1^2).
%p A141215 select(p -> isprime(p) and nops([isolve(x^2 - 61*y^2 = p)])>0, [seq(2*i+1,i=1..1000)]); # _Robert Israel_, Jun 11 2014
%t A141215 terms = 100; d = 61;
%t A141215 Table[3*x^2 + 5*x*y - 3*y^2, {x, 1, terms}, {y, Floor[(5 - Sqrt[d])*x/6], Ceiling[(5 + Sqrt[d])*x/6]}] // Flatten // Select[#, Positive[#] && PrimeQ[#]&]& // Union // Take[#, terms]& (* _Jean-François Alcover_, Feb 28 2019 *)
%Y A141215 Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
%Y A141215 Primes in A243654.
%Y A141215 For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
%K A141215 nonn
%O A141215 1,1
%A A141215 Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 14 2008