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A141158 Primes of the form x^2 + 4*x*y - y^2. 19
5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Discriminant = 20. 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.

Values of the quadratic form are {0,1,4} mod 5, so this is a subset of A038872. - R. J. Mathar, Jul 30 2008

Is this the same sequence as A038872?

Also primes of the form u^2-5v^2. The transformation {u,v}={x+2y,y} transforms it into the one in the title. - Tito Piezas III, Dec 28 2008

REFERENCES

Borevich and Shafaewich, Number Theory.

D. B. Zagier, Zetafunktionen und quadratische Koerper.

LINKS

Table of n, a(n) for n=1..54.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

EXAMPLE

a(3)=19 because we can write 19=2^2+4*2*5-5^2.

MATHEMATICA

lim = 25; Select[Union[Flatten[Table[x^2 + 4 x y - y^2, {x, 0, lim}, {y, 0, lim}]]], # > 0 && # < lim^2 && PrimeQ[#] &] (* T. D. Noe, Aug 31 2012 *)

CROSSREFS

Cf. A038872 (Primes congruent to {0, 1, 4} mod 5. ) A038872 (d=5). A038873 (d=8), A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).

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: A089270 A275068 A038872 * A239732 A130828 A244241

Adjacent sequences:  A141155 A141156 A141157 * A141159 A141160 A141161

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 12 2008

STATUS

approved

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Last modified September 21 15:13 EDT 2017. Contains 292300 sequences.