A325086 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 448*y^2.

457, 569, 617, 809, 1289, 1801, 1913, 2153, 2297, 2473, 2521, 2633, 3049, 3257, 3929, 4057, 4153, 4201, 4937, 5209, 5273, 5881, 6073, 6553, 6841, 7177, 7193, 7417, 7529, 7673, 7753, 8009, 8521, 8537, 8681, 9769, 10889, 11257, 11321, 11369, 11593, 11657, 11897

Offset

1,1

Comments

Brink showed that prime numbers congruent to 9, 25 or 57 modulo 112 are representable by exactly one of the quadratic forms x^2 + 14*y^2 or x^2 + 448*y^2. A325085 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

Links

Table of n, a(n) for n=1..43.

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 A325086

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 7177:
- 7177 is a prime number,
- 7177 = 64*112 + 9,
- 7177 = 3^2 + 448*4^2,
- hence 7177 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325085.

Sequence in context: A068259 A036271 A061326 * A215895 A142828 A020364

Adjacent sequences: A325083 A325084 A325085 * A325087 A325088 A325089

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved