

A139827


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


253



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
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OFFSET

1,1


COMMENTS

Discriminant = 132.
Consider the quadratic form f(x,y) = ax^2 + bxy + cy^2. When the discriminant d=b^24ac 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 ith 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

T. D. Noe and 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

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 *)


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, A139841A139843 (d=408), A139644, A139844A139850 (d=420), A139645, A139851A139853 (d=448), A139502, A139854A139860 (d=480), A139646, A139861A139863 (d=520), A139647, A139864A139866 (d=532), A139648, A139867A139873 (d=660), A139506, A139874A139880 (d=672), A139649, A139881A139883 (d=708), A139650, A139884A139886 (d=760), A139651, A139887A139893 (d=840), A139652, A139894A139896 (d=928), A139502, A139855, A139857, A139858, A139897A139899, A139902 (d=960).
Cf. also A139653, A139904A139906 (d=1012), A139654, A139907A139913 (d=1092), A139655, A139914A139920 (d=1120), A139656, A139921A139927 (d=1248), A139657, A139928A139934 (d=1320), A139658, A139935A139941 (d=1380), A139659, A139942A139948 (d=1428), A139660, A139949A139955 (d=1540), A139661, A139956A139962 (d=1632), A139662, A139963A139969 (d=1848), A139663, A139970A139976 (d=2080), A139664, A139977A139983 (d=3040), A139665, A139984A139998 (d=3360), A139666, A139999A140013 (d=5280), A139667, A140014A140028 (d=5460), A139668, A140029A140043 (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.
Sequence in context: A171605 A018759 A132146 * A197186 A063118 A267540
Adjacent sequences: A139824 A139825 A139826 * A139828 A139829 A139830


KEYWORD

nonn,easy


AUTHOR

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


STATUS

approved



