A325082 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + x*y + 69*y^2.

89, 179, 419, 449, 599, 619, 709, 719, 829, 859, 1039, 1109, 1259, 1489, 1549, 1709, 1879, 2039, 2099, 2179, 2539, 2579, 2689, 2909, 3169, 3259, 3359, 3389, 3499, 3919, 4019, 4159, 4229, 4349, 4409, 4799, 4909, 5009, 5039, 5179, 5449, 5569, 5659, 5779, 5839

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. A325081 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

Examples

			Regarding 2099:
- 2099 is a prime number,
- 2099 = 38*55 + 9,
- 2099 = 17^2 + 1*17*5 + 69*5^2,
- hence 2099 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 A325082
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325081.

Programs

PARI

See Links section.