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A141161 Primes of the form 4*x^2+6*x*y-7*y^2. 8
3, 7, 11, 41, 47, 53, 71, 73, 83, 101, 127, 149, 157, 173, 181, 197, 211, 223, 229, 263, 271, 307, 337, 359, 373, 379, 397, 419, 433, 443, 509, 521, 571, 593, 599, 613, 617, 619, 641, 659, 673, 677, 719, 733, 739, 743, 751, 761, 773, 787, 811, 821, 887, 937 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Discriminant = 148. Class = 3. 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 represented by the improperly equivalent form 7*x^2 + 6*x*y - 4*y^2
REFERENCES
Z. I. Borevich and I. R. Shafarevich, Number Theory.
LINKS
Peter Luschny, Binary Quadratic Forms
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(8)=73 because we can write 73= 4*4^2+6*4*3-7*3^2.
MATHEMATICA
q := 4*x^2 + 6*x*y - 7*y^2; pmax = 1000; xmax = 100; ymin = -xmax; ymax = xmax; k = 1; 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, k*xmax}, {y, k*ymin, k*ymax}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; k++; Print["k = ", k, " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141161 = prms (* Jean-François Alcover, Oct 26 2016 *)
PROG
(Sage)
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([4, 6, -7])
Q.represented_positives(937, '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). A141163 (d=148).
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: A106966 A191027 A139599 * A333421 A358311 A217383
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 13 2008
STATUS
approved

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Last modified May 29 10:24 EDT 2024. Contains 372938 sequences. (Running on oeis4.)