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).

3, 5, 13, 19, 41, 47, 61, 73, 83, 97, 103, 107, 109, 113, 127, 131, 137, 149, 163, 167, 179, 197, 199, 229, 239, 241, 257, 263, 269, 271, 283, 293, 317, 347, 353, 367, 379, 431, 439, 443, 449, 461, 463, 479, 487, 491, 503, 563, 569, 571, 601, 607, 613, 619

Offset: 1

Author

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

Keywords

• nonn

Comments

Discriminant = 61. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

A subsequence of (and may possibly coincide with) A038941. - R. J. Mathar, Jul 22 2008

3*x^2+5*x*y-3*y^2 and 5*x^2+9*x*y+y^2 are equivalent forms.

Also, primes of the form x^2 - 61y^2, of discriminant 244.

Examples

			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).
		

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Crossrefs

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

Primes in A243654.

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

Programs

Maple

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

Mathematica

terms = 100; d = 61;
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 *)