A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2.

17, 97, 193, 241, 401, 433, 449, 641, 673, 769, 929, 977, 1009, 1297, 1361, 1409, 1489, 1697, 1873, 2017, 2081, 2161, 2417, 2609, 2753, 2801, 2897, 3041, 3169, 3329, 3457, 3617, 3697, 3793, 3889, 4129, 4241, 4337, 4561, 4673, 5009, 5153, 5281, 5441, 5521, 5857

Offset: 1

Author

Rémy Sigrist, Mar 27 2019

Keywords

• nonn

Comments

Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. A325067 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

Examples

			Regarding 17:
- 17 is a prime number,
- 17 = 16*1 + 1,
- 17 is representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2,
- hence 17 belongs to the 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 A325068
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

Cf. A094407, A325067.

Programs

PARI

See Links section.