A325078 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + x*y + 127*y^2.

127, 199, 283, 337, 433, 571, 727, 829, 883, 907, 1213, 1291, 1297, 1447, 1531, 1609, 1663, 1741, 2053, 2383, 2389, 2677, 3169, 3301, 3319, 3631, 3691, 3709, 3769, 3793, 4003, 4099, 4159, 4549, 4567, 4651, 4729, 4801, 4957, 5347, 5407, 5431, 5563, 5821, 6133

Offset

1,1

Comments

Brink showed that prime numbers congruent to 4, 10 or 25 modulo 39 are representable by exactly one of the quadratic forms x^2 + x*y + 10*y^2 or x^2 + x*y + 127*y^2. A325077 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

Links

Table of n, a(n) for n=1..45.

David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.

Rémy Sigrist, PARI program for A325078

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 127:
- 127 is a prime number,
- 127 = 3*39 + 10,
- 127 = 0^2 + 0*1 + 127*1^2,
- hence 127 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325077.

Sequence in context: A195377 A142090 A095730 * A276261 A343319 A045117

Adjacent sequences: A325075 A325076 A325077 * A325079 A325080 A325081

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved