OFFSET
1,1
COMMENTS
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.
REFERENCES
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
LINKS
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)
FORMULA
The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).
MATHEMATICA
PROG
(Magma) [ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // Vincenzo Librandi, Jul 29 2012
(PARI) v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v, p%132), primes(100)) \\ Charles R Greathouse IV, Jan 08 2013
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
T. D. Noe, May 02 2008, May 07 2008
STATUS
approved