A141166 Primes of the form x^2+15*x*y-y^2.

37, 53, 173, 193, 229, 241, 347, 359, 383, 439, 443, 449, 461, 503, 509, 541, 593, 607, 617, 619, 643, 691, 907, 967, 977, 1019, 1051, 1063, 1097, 1109, 1249, 1277, 1291, 1303, 1321, 1399, 1429, 1583, 1667, 1741, 1783, 1993, 1997, 2003, 2087, 2137, 2143, 2333, 2347, 2351

Offset

1,1

Comments

Discriminant = 229. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d = b^2-4ac. They can represent primes only if gcd(a,b,c)=1. [Edited by M. F. Hasler, Jan 27 2016]

Appears to be the complement of A141165 in A268155, primes that are squares mod 229. - M. F. Hasler, Jan 27 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(2)=53 because we can write 53= 3^2+15*3*1-1^2

Mathematica

lim = 100; Rest@ Union@ Abs@ Flatten@ Table[x^2 + 15 x y - y^2, {x, lim}, {y, lim}] /. n_ /; CompositeQ@ n -> Nothing (* Michael De Vlieger, Jan 27 2016 *)

Prog

(PARI) is_A141166(p)=qfbsolve(Qfb(1, 15, -1), p) \\ Returns nonzero (actually, a solution [x, y]) iff p is a member of the sequence. For efficiency it is assumed that p is prime. Example usage: select(is_A141166, primes(500)) - M. F. Hasler, Jan 27 2016

Crossrefs

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141165 (d=229).

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: A330339 A036540 A225214 * A242930 A139918 A289510

Adjacent sequences: A141163 A141164 A141165 * A141167 A141168 A141169

Keyword

nonn

Author

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

Status

approved