|
|
A141178
|
|
Primes of the form 3*x^2+x*y-3*y^2 (as well as of the form 3*x^2+7*x*y+y^2).
|
|
6
|
|
|
3, 7, 11, 37, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149, 151, 157, 173, 181, 197, 211, 223, 229, 233, 263, 269, 271, 293, 307, 317, 337, 349, 359, 367, 373, 379, 397, 419, 433, 443, 491, 509, 521, 571, 593, 599, 601, 613, 617, 619, 641, 659, 673
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Discriminant = 37. 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
|
|
REFERENCES
|
Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
|
|
LINKS
|
|
|
EXAMPLE
|
a(3) = 11 because we can write 11 = 3*2^2+2*1-3*1^2 (or 11 = 3*1^2+7*1*1+1^2).
|
|
MATHEMATICA
|
Reap[For[p = 2, p < 1000, p = NextPrime[p], If[FindInstance[p == 3*x^2 + x*y - 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 37 || MatchQ[Mod[#, 37], Alternatives[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]]&](* Jean-François Alcover, Oct 25 2016, updated Oct 30 2016 *)
|
|
CROSSREFS
|
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008
|
|
STATUS
|
approved
|
|
|
|