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A141172 Primes of the form 2*x^2+2*x*y-3*y^2 (as well as of the form 2*x^2+6*x*y+y^2). 7
2, 29, 37, 53, 109, 113, 137, 149, 193, 197, 233, 277, 281, 317, 337, 373, 389, 401, 421, 449, 457, 541, 557, 569, 613, 617, 641, 653, 673, 701, 709, 757, 809, 821, 877, 953, 977, 1009, 1033, 1061, 1093, 1117, 1129, 1201, 1213, 1229, 1289, 1297, 1373, 1381, 1409, 1429, 1453, 1481, 1493 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Discriminant = 28. 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.

Also, primes of form u^2-7v^2. The transformation {u,v}={3x+y,x} yields the second quadratic form given in the title. - Tito Piezas III, Dec 28 2008

This is also the list of primes p such that p = 2 or p is congruent to 1, 9 or 25 mod 28 - Jean-François Alcover, Oct 28 2016

REFERENCES

Borevich and Shafaewich, Number Theory.

D. B. Zagier, Zetafunktionen und quadratische Koerper.

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)

EXAMPLE

a(2)=29 because we can write 29=2*4^2+2*4*3-3*3^2 (or 29=2*1^2+6*1*3+3^2).

MATHEMATICA

Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 28], 1|9|25]&] (* Jean-François Alcover, Oct 28 2016 *)

CROSSREFS

Cf. A141173 (d=28) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).

Cf. also A242662.

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: A019392 A060503 A180231 * A285688 A139833 A260792

Adjacent sequences:  A141169 A141170 A141171 * A141173 A141174 A141175

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified June 17 04:26 EDT 2019. Contains 324183 sequences. (Running on oeis4.)