

A141158


Primes of the form x^2 + 4*x*y  y^2.


20



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
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OFFSET

1,1


COMMENTS

Discriminant = 20. Class number = 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 (primitive).
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? [Yes. See a comment in A038872, and the comment by Jianing Song below.  Wolfdieter Lang, Jun 19 2019]
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
From Jianing Song, Sep 20 2018: (Start)
Yes, this is a duplicate of A038872. For primes p congruent to {1, 4} mod 5, they split in the quadratic field Q[sqrt(5)]. Since Q[sqrt(5)] is a UFD, they are reducible in Q[sqrt(5)], so we have p = (u + v*sqrt(5))*(u  v*sqrt(5)) = u^2  5*v^2. On the other hand, u^2  5*v^2 == 0, 1, 4 (mod 5). So these two sequences are the same.
Also primes of the form x^2  x*y  y^2 (discriminant 5) with 0 <= x <= y (or x^2 + x*y  y^2 with x, y nonnegative).
(End)


REFERENCES

Z. I. Borevich and I. R. Shafarevich, 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 (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



