A141303 - OEIS

A141303

Primes of the form 2*x^2+6*x*y-3*y^2 (as well as of the form 5*x^2+10*x*y+2*y^2).

2, 5, 17, 53, 113, 137, 173, 197, 233, 257, 293, 317, 353, 557, 593, 617, 653, 677, 773, 797, 857, 953, 977, 1013, 1097, 1193, 1217, 1277, 1373, 1433, 1493, 1553, 1613, 1637, 1697, 1733, 1877, 1913, 1973, 1997, 2153, 2213, 2237, 2273, 2297, 2333, 2357, 2393, 2417, 2477

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OFFSET

1,1

COMMENTS

Discriminant = 60. Class = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

This is also the list of primes p such that p = 2 or 5 or p is congruent to 17 or 53 mod 60. - Jean-François Alcover, Oct 28 2016

REFERENCES

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

LINKS

Juan Arias-de-Reyna, 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.

EXAMPLE

a(3)=17 because we can write 17=2*2^2+6*2*1-3*1^2 (or 17=5*1^2+10*1*1+2*1^2).

MATHEMATICA

Select[Prime[Range[500]], # == 2 || # == 5 || MatchQ[Mod[#, 60], 17|53]&] (* Jean-François Alcover, Oct 28 2016 *)

CROSSREFS

Cf. A107152, A141302, A141304 (d=60), A107167.

Primes in A243189.

Sequence in context: A216969 A148406 A148407 * A263678 A133510 A191640

Adjacent sequences: A141300 A141301 A141302 * A141304 A141305 A141306

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