A325085 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 14*y^2.

137, 233, 281, 953, 1033, 1129, 1481, 2137, 2377, 2713, 2857, 2969, 3529, 3593, 3833, 4649, 4729, 5657, 5737, 5849, 6217, 6329, 6521, 6857, 7001, 7561, 8089, 8233, 8297, 8761, 8969, 9209, 9241, 9433, 9689, 10313, 11113, 12377, 12457, 12553, 12601, 12713, 12889

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. This sequence corresponds to those representable by the first form, and A325086 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 A325085

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 11113:
- 11113 is a prime number,
- 11113 = 99*112 + 25,
- 11113 = 103^2 + 14*6^2,
- hence 11113 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325086.

Sequence in context: A142257 A141926 A107164 * A142497 A142523 A307839

Adjacent sequences: A325082 A325083 A325084 * A325086 A325087 A325088

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved