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A141193 Primes of the form -3*x^2+3*x*y+4*y^2 (as well as of the form 6*x^2+9*x*y+y^2). 7
7, 19, 43, 61, 73, 139, 157, 163, 199, 229, 271, 277, 283, 313, 349, 367, 397, 457, 463, 499, 541, 571, 577, 613, 619, 631, 643, 691, 709, 727, 733, 739, 757, 769, 823, 853, 859, 883, 919, 937, 967, 997, 1033, 1051, 1069, 1087, 1201, 1213, 1279, 1297, 1303, 1327, 1423, 1429 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Discriminant = 57. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1
p = 19 and primes p = 1 mod 3 such that 57 is a square mod p. - Juan Arias-de-Reyna, Mar 20 2011
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
EXAMPLE
a(2)=19 because we can write 19=-3*1^2+3*1*2+4*2^2
MATHEMATICA
Select[Prime[Range[250]], # == 19 || MatchQ[Mod[#, 57], Alternatives[1, 4, 7, 16, 25, 28, 43, 49, 55]]&] (* Jean-François Alcover, Oct 28 2016 *)
CROSSREFS
Cf. A141192 (d=57). A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243193.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Sequence in context: A002177 A225279 A192755 * A104163 A145993 A265676
KEYWORD
nonn
AUTHOR
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
STATUS
approved

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Last modified February 29 21:35 EST 2024. Contains 370428 sequences. (Running on oeis4.)