A325080 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 neither representable by x^2 + x*y + 14*y^2 nor by x^2 + x*y + 69*y^2.

31, 181, 191, 331, 401, 421, 521, 641, 911, 971, 991, 1021, 1291, 1301, 1511, 1621, 1831, 1871, 2011, 2161, 2281, 2311, 2381, 2861, 3001, 3041, 3061, 3221, 3301, 3331, 3391, 3821, 3931, 4051, 4211, 4261, 4271, 4621, 4691, 4801, 4871, 4931, 4951, 5021, 5171

Offset: 1

Author

Rémy Sigrist, Mar 28 2019

Keywords

• nonn

Comments

Brink showed that prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 are representable by both or neither of the quadratic forms x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2. A325079 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

Examples

			Regarding 31:
- 31 is a prime number,
- 31 = 0*55 + 31,
- 31 is neither representable by x^2 + x*y + 14*y^2 nor by x^2 + x*y + 69*y^2,
- hence 31 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 A325080
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325079.

Programs

PARI

See Links section.