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A141165
Primes of the form 9*x^2+7*x*y-5*y^2.
8
3, 5, 11, 17, 19, 43, 61, 71, 83, 97, 103, 149, 151, 167, 181, 233, 271, 277, 293, 307, 311, 337, 367, 373, 397, 401, 409, 421, 431, 433, 457, 463, 467, 491, 557, 569, 587, 631, 641, 661, 673, 683, 701, 733, 743, 751, 757, 769, 787, 821, 859, 863, 883, 911
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]
Also primes represented by the improperly equivalent form 5*x^2+7*x*y-9*y^2. - Juan Arias-de-Reyna, Mar 17 2011
36*a(n) has the form z^2 - 229*y^2, where z = 18*x+7*y. [Bruno Berselli, Jun 25 2014]
Appears to be the complement of A141166 in A268155, primes that are squares mod 229. - M. F. Hasler, Jan 27 2016
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory
D. B. Zagier, Zetafunktionen und quadratische Körper
LINKS
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
EXAMPLE
a(10)=97 because we can write 97= 9*3^2+7*3*1-5*1^2
MATHEMATICA
q := 9*x^2 + 7*x*y - 5*y^2; pmax = 1000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (expansion coeff. for maxima *); prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, If[xmax == xmax0, xmax, Floor[k*xmax]]}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141165 = prms (* Jean-François Alcover, Oct 26 2016 *)
PROG
(PARI) is_A141165(p)=qfbsolve(Qfb(9, 7, -5), 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. - M. F. Hasler, Jan 27 2016
(Sage) # uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([9, 7, -5])
print(Q.represented_positives(911, 'prime')) # Peter Luschny, Oct 26 2016
CROSSREFS
Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141166 (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: A191141 A268155 A199217 * A220954 A155937 A050566
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