A139665 -id:A139665 - cp's OEIS frontend

A139827 Primes of the form 2x^2 + 2xy + 17y^2.

2, 17, 29, 41, 101, 149, 173, 197, 233, 281, 293, 461, 557, 569, 593, 677, 701, 761, 809, 821, 857, 941, 953, 1097, 1217, 1229, 1289, 1361, 1481, 1493, 1553, 1601, 1613, 1733, 1877, 1889, 1913, 1949, 1997, 2081, 2129, 2141, 2153, 2213, 2273, 2309, 2393, 2417

Offset: 1

T. D. Noe, May 02 2008, May 07 2008

Discriminant = -132.
Consider the quadratic form f(x,y) = ax^2 + bxy + cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in the OEIS.
When a=1 and b=0, f(x,y) is a quadratic form whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such that the Jacobi symbol (-k/4N)=1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A139841-A139843 (d=-408), A139644, A139844-A139850 (d=-420), A139645, A139851-A139853 (d=-448), A139502, A139854-A139860 (d=-480), A139646, A139861-A139863 (d=-520), A139647, A139864-A139866 (d=-532), A139648, A139867-A139873 (d=-660), A139506, A139874-A139880 (d=-672), A139649, A139881-A139883 (d=-708), A139650, A139884-A139886 (d=-760), A139651, A139887-A139893 (d=-840), A139652, A139894-A139896 (d=-928), A139502, A139855, A139857, A139858, A139897-A139899, A139902 (d=-960).
Cf. also  A139653, A139904-A139906 (d=-1012), A139654, A139907-A139913 (d=-1092), A139655, A139914-A139920 (d=-1120), A139656, A139921-A139927 (d=-1248), A139657, A139928-A139934 (d=-1320), A139658, A139935-A139941 (d=-1380), A139659, A139942-A139948 (d=-1428), A139660, A139949-A139955 (d=-1540), A139661, A139956-A139962 (d=-1632), A139662, A139963-A139969 (d=-1848), A139663, A139970-A139976 (d=-2080), A139664, A139977-A139983 (d=-3040), A139665, A139984-A139998 (d=-3360), A139666, A139999-A140013 (d=-5280), A139667, A140014-A140028 (d=-5460), A139668, A140029-A140043 (d=-7392).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[2, -2, 17, 2500] (* see A106856 *)
t = Table[{2, 17, 29, 41, 65, 101} + 132*n, {n, 0, 50}]; Select[Flatten[t], PrimeQ] (* T. D. Noe, Jun 21 2012 *)

v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v,p%132),primes(100)) \\ Charles R Greathouse IV, Jan 08 2013

The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).

A173274 Primes of the form x^2 + 18480*y^2.

Original entry on oeis.org

18481, 19009, 19441, 20161, 21961, 31249, 41281, 47041, 48409, 51241, 68209, 70009, 70921, 74209, 74449, 74761, 75289, 76129, 76561, 77641, 80809, 84121, 85369, 86689, 87649, 90841, 91081, 91921, 93241, 97441, 102001, 102481, 106681

Offset: 1

Michel Lagneau, Feb 14 2010, Jun 08 2010

nonn

The primes p of the form x^2 + 18480*y^2 are also of the multi-forms x^2 + y^2, x^2 + 2*y^2, x^2 + 3*y^2, ..., x^2 + 11*y^2, x^2 + 12*y^2, but the reverse is false. For example, p = 7561 has twelve forms, but is not of the form x^2 + 18480*y^2.

			18481 = 1^2 + 18480*1^2 and also 18481 = 16^2 + 135^2 = 7^2 + 2*96^2 = 127^2 + 3*28^2 = 135^2 + 4*8^2 = 74^2 + 5*51^2 = 59^2 + 6*50^2 = 97^2 + 7*36^2 = 7^2 + 8*48^2 = 16^2 + 9*45^2 = 29^2 + 10*42^2 = 65^2 + 11*36^2 = 127^2 + 12*14^2.

David A. Cox, "Primes of the Form x^2 + n*y^2", Wiley, 1989, Section 3.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1848, p. 146, Ellipses, Paris 2008.

Ray Chandler, 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)
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

Cf. A139668: primes of the form x^2 + 1848*y^2;

Cf. A139665: primes of the form x^2 + 840*y^2.

fd:=proc(a,b,c,M) local dd,xlim,ylim,x,y,t1,t2,t3,t4,i;
dd:=4*a*c-b^2;
if dd<=0 then error "Form should be positive definite."; break; fi;
t1:={};
xlim:=ceil( sqrt(M/a)*(1+abs(b)/sqrt(dd)));
ylim:=ceil( 2*sqrt(a*M/dd));
for x from 0 to xlim do
for y from -ylim to ylim do
t2 := a*x^2+b*x*y+c*y^2;
if t2 <= M then t1:={op(t1),t2}; fi; od: od:
t3:=sort(convert(t1,list));
t4:=[];
for i from 1 to nops(t3) do
   if isprime(t3[i]) then t4:=[op(t4),t3[i]]; fi; od:
[[seq(t3[i],i=1..nops(t3))], [seq(t4[i],i=1..nops(t4))]];
end;
fd(1,0,18480,100000);

QuadPrimes2[1, 0, 18480, 100000] (* see A106856 *)
(* Second program: *)
max = 107000; m = 18480; Table[yy = {y, 1, Floor[Sqrt[max-x^2]/(Sqrt[m])]}; Table[x^2 + m y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]&

fc(a,b,c,M) = {
  my(t1=List(),t2);
  forprime(p=2,prime(M),
    t2 = qfbsolve(Qfb(a,b,c),p);
    if(t2 != 0, listput(t1,p))
  );
  Vec(t1)
};
fc(1,0,18480,100000)

Corrected sequence and replaced defective program. - Ray Chandler, Aug 14 2014

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A139827 Primes of the form 2x^2 + 2xy + 17y^2.

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