%I #42 Oct 30 2023 07:37:43
%S 2,17,29,41,101,149,173,197,233,281,293,461,557,569,593,677,701,761,
%T 809,821,857,941,953,1097,1217,1229,1289,1361,1481,1493,1553,1601,
%U 1613,1733,1877,1889,1913,1949,1997,2081,2129,2141,2153,2213,2273,2309,2393,2417
%N Primes of the form 2x^2 + 2xy + 17y^2.
%C Discriminant = -132.
%C 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.
%C 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.
%D David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
%H Ray Chandler, <a href="/A139827/b139827.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%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)
%F The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).
%t QuadPrimes2[2, -2, 17, 2500] (* see A106856 *)
%t t = Table[{2, 17, 29, 41, 65, 101} + 132*n, {n, 0, 50}]; Select[Flatten[t], PrimeQ] (* _T. D. Noe_, Jun 21 2012 *)
%o (Magma) [ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // _Vincenzo Librandi_, Jul 29 2012
%o (PARI) v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v,p%132),primes(100)) \\ _Charles R Greathouse IV_, Jan 08 2013
%Y 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).
%Y 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).
%Y 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.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, May 02 2008, May 07 2008