A141165 - OEIS

%I #38 Feb 29 2020 18:10:26

%S 3,5,11,17,19,43,61,71,83,97,103,149,151,167,181,233,271,277,293,307,

%T 311,337,367,373,397,401,409,421,431,433,457,463,467,491,557,569,587,

%U 631,641,661,673,683,701,733,743,751,757,769,787,821,859,863,883,911

%N Primes of the form 9*x^2+7*x*y-5*y^2.

%C 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]

%C Also primes represented by the improperly equivalent form 5*x^2+7*x*y-9*y^2. - _Juan Arias-de-Reyna_, Mar 17 2011

%C 36*a(n) has the form z^2 - 229*y^2, where z = 18*x+7*y. [_Bruno Berselli_, Jun 25 2014]

%C Appears to be the complement of A141166 in A268155, primes that are squares mod 229. - _M. F. Hasler_, Jan 27 2016

%D Z. I. Borevich and I. R. Shafarevich, Number Theory

%D D. B. Zagier, Zetafunktionen und quadratische Körper

%H Juan Arias-de-Reyna, <a href="/A141165/b141165.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms#Implementation">Binary Quadratic Forms</a>

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%e a(10)=97 because we can write 97= 9*3^2+7*3*1-5*1^2

%t 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 *)

%o (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

%o (Sage) # uses[binaryQF]

%o # The function binaryQF is defined in the link 'Binary Quadratic Forms'.

%o Q = binaryQF([9, 7, -5])

%o print(Q.represented_positives(911, 'prime')) # _Peter Luschny_, Oct 26 2016

%Y Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141166 (d=229).

%Y For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

%K nonn

%O 1,1

%A 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