A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32*y^2 and x^2 + 64*y^2.

113, 257, 337, 353, 577, 593, 881, 1153, 1201, 1217, 1249, 1553, 1601, 1777, 1889, 2113, 2129, 2273, 2593, 2657, 2689, 2833, 3089, 3121, 3137, 3217, 3313, 3361, 3761, 4001, 4049, 4177, 4273, 4289, 4481, 4513, 4657, 4721, 4801, 4817, 4993, 5233, 5297, 5393

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. This sequence corresponds to those representable by both, and A325068 corresponds to those representable by neither.

Also, Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of these quadratic forms. A325069 corresponds to those representable by the first form and A325070 to those representable by the second form.

Brink provided similar results for other congruences.

Examples

			Regarding 1201:
- 1201 is a prime number,
- 1201 = 75*16 + 1,
- 1201 = 7^2 + 32*6^2 = 25^2 + 64*3^2,
- hence 1201 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 A325067
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

Cf. A094407, A105126, A325068, A325069, A325070.

See A325071, A325072, A325073 and A325074 for similar results in congruences modulo 16.

See A325075, A325076, A325077 and A325078 for similar results in congruences modulo 39.

See A325079, A325080, A325081 and A325082 for similar results in congruences modulo 55.

See A325083, A325084, A325085 and A325086 for similar results in congruences modulo 112.

See A325087, A325088, A325089 and A325090 for similar results in congruences modulo 240.

Programs

PARI

See Links section.