A325084 Prime numbers congruent to 1, 65 or 81 modulo 112 neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2.

113, 193, 337, 401, 641, 1009, 1201, 1297, 2689, 2801, 3089, 3137, 3217, 3329, 3361, 3761, 3889, 4337, 4481, 5009, 5153, 5233, 5441, 5569, 6113, 6337, 6353, 6449, 6577, 6673, 7681, 7841, 8513, 8737, 8929, 9041, 9137, 9521, 9601, 9697, 10369, 10529, 10753

Offset

1,1

Comments

Brink showed that prime numbers congruent to 1, 65 or 81 modulo 112 are representable by both or neither of the quadratic forms x^2 + 14*y^2 and x^2 + 448*y^2. A325083 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

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 A325084

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 113:
- 113 is a prime number,
- 113 = 1*112 + 1,
- 113 is neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2,
- hence 113 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325083.

Sequence in context: A142303 A152929 A142180 * A084951 A151947 A087703

Adjacent sequences: A325081 A325082 A325083 * A325085 A325086 A325087

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved