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

1,1

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.

Links

Table of n, a(n) for n=1..44.

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

Example

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.

Prog

(PARI) See Links section.

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.

Sequence in context: A142002 A142917 A142947 * A142641 A070181 A001134

Adjacent sequences: A325064 A325065 A325066 * A325068 A325069 A325070

Keyword

nonn

Author

Rémy Sigrist, Mar 27 2019

Status

approved