1,1

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 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 the Jacobi symbol (-k/4N)=1.

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

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

f[x_, y_]:=2*x^2+2*x*y+17*y^2; lst={}; Do[Do[p=f[x, y]; If[PrimeQ[p], AppendTo[lst, p]], {y, -4!, 3*4!}], {x, -4!, 3*4!}]; Take[Union[lst], 90] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009]

QuadPrimes[2, -2, 17, 10000] (* see A106856 *)

Cf. A002313 (d=-4) and the following:

A033200 (d=-8),

A007645 (d=-12),

A002144 (d=-16),

A033205, A106865 (d=-20),

A033199, A084865 (d=-24),

A033207 (d=-28),

A007519, A007520 (d=-32),

A068228, A040117 (d=-36),

A033201, A106889 (d=-40),

A068228, A068229 (d=-48),

A033210, A106906 (d=-52),

A033212, A106859 (d=-60),

A007519, A007521 (d=-64),

A106950, A106949 (d=-72),

A033215, A102271, A102273, A106972 (d=-84),

A033216, A106984 (d=-88),

A107008, A107003, A107006, A107007 (d=-96),

A033205, A122487 (d=-100),

A107134, A107133 (d=-112),

A033220, A107135, A107136, A107137 (d=-120),

A033222, A107138, A139827, A139828 (d=-132),

A033225, A007639 (d=-148),

A107145, A107144, A139829, A139830 (d=-160),

A033229, A107146, A107147, A107148 (d=-168),

A107152, A107151, A139831, A139832 (d=-180),

A107008, A107006, A107154, A139530 (d=-192),

A033236, A107165, A139833, A139834 (d=-228),

A033237, A107166 (d=-232),

A107152, A107167, A107168, A107169 (d=-240),

A033245, A107178, A107179, A107180 (d=-280),

A107008, A107007, A107154, A107181 (d=-288),

A033250, A107188, A107189, A107190 (d=-312),

A033254, A107199, A139835, A139836 (d=-340),

A107202, A107201, A139837, A139838 (d=-352),

A033202, A107210, A139839, A139840 (d=-372),

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),

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).

Sequence in context: A164275 A018759 A132146 this_sequence A063118 A141068 A162622

Adjacent sequences: A139824 A139825 A139826 this_sequence A139828 A139829 A139830

nonn,easy

T. D. Noe (noe(AT)sspectra.com), May 02 2008, May 07 2008