login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000

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

Adjacent sequences:  A141162 A141163 A141164 * A141166 A141167 A141168

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 28 18:04 EDT 2021. Contains 347716 sequences. (Running on oeis4.)