A325070 Prime numbers congruent to 9 modulo 16 representable by x^2 + 64*y^2.

73, 89, 233, 281, 601, 617, 937, 1033, 1049, 1097, 1193, 1289, 1433, 1481, 1609, 1721, 1753, 1801, 1913, 2089, 2281, 2393, 2441, 2473, 2857, 2969, 3049, 3257, 3449, 3529, 3673, 3833, 4057, 4153, 4201, 4217, 4297, 4409, 4457, 4937, 5081, 5113, 5209, 5689, 5737

Offset

1,1

Comments

Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of the quadratic forms x^2 + 32*y^2 or x^2 + 64*y^2. A325069 corresponds to those representable by the first form and this sequence to those representable by the second form.

Links

Table of n, a(n) for n=1..45.

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 A325070

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 4201:
- 4201 is a prime number,
- 4201 = 262*16 + 9,
- 4201 = 51^2 + 64*5^2,
- hence 4201 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A105126, A325069.

Sequence in context: A007766 A365001 A065111 * A152308 A072052 A333391

Adjacent sequences: A325067 A325068 A325069 * A325071 A325072 A325073

Keyword

nonn

Author

Rémy Sigrist, Mar 27 2019

Status

approved