A325083 Prime numbers congruent to 1, 65 or 81 modulo 112 representable by both x^2 + 14*y^2 and x^2 + 448*y^2.

449, 673, 977, 1409, 1873, 2017, 2081, 2129, 2417, 2657, 2753, 3313, 3697, 4001, 4561, 4657, 4673, 4817, 4993, 6689, 6833, 7057, 7121, 7393, 7457, 7793, 8017, 8353, 8369, 8689, 8849, 9377, 9473, 9857, 10193, 10273, 11057, 11393, 11489, 11953, 12161, 12289

Offset: 1

Author

Rémy Sigrist, Mar 28 2019

Keywords

• nonn

Comments

Brink showed that prime numbers congruent to 1, 65 or 81 modulo 112 are representable by both or neither of the quadratic forms x^2 + 14*y^2 and x^2 + 448*y^2. This sequence corresponds to those representable by both, and A325084 corresponds to those representable by neither.

Examples

			Regarding 3313:
- 3313 is a prime number,
- 3313 = 29*112 + 65,
- 3313 = 53^2 + 14*6^2 = 39^2 + 448*2^2,
- hence 3313 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 A325083
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325084.

Programs

PARI

See Links section.