A325081 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + x*y + 14*y^2.

59, 199, 229, 269, 379, 389, 499, 509, 839, 929, 1049, 1279, 1409, 1439, 1499, 1609, 1699, 2029, 2069, 2269, 2399, 2699, 2729, 2819, 3019, 3089, 3469, 3529, 3719, 4049, 4079, 4129, 4139, 4339, 4519, 4679, 4789, 4889, 4999, 5119, 5399, 5479, 5669, 6029, 6229

Offset: 1

Author

Rémy Sigrist, Mar 28 2019

Keywords

• nonn

Comments

Brink showed that prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 are representable by exactly one of the quadratic forms x^2 + x*y + 14*y^2 or x^2 + x*y + 69*y^2. This sequence corresponds to those representable by the first form, and A325082 corresponds to those representable by the second form.

Examples

			Regarding 4999:
- 4999 is a prime number,
- 4999 = 90*55 + 49,
- 4999 = 41^2 + 41*14 + 14*14^2,
- hence 4999 belongs to this sequence.
		

Links

• 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 A325081
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325082.

Programs

PARI

See Links section.