A325079 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 representable by both x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2.

71, 251, 311, 631, 661, 691, 751, 881, 1061, 1171, 1181, 1321, 1571, 1721, 1741, 1901, 1951, 2341, 2531, 2621, 2671, 2711, 2731, 2971, 3191, 3271, 3371, 3491, 3631, 3701, 3851, 3881, 4481, 4591, 4651, 5261, 5471, 5501, 5531, 5581, 5641, 5701, 5861, 6121, 6271

Offset

1,1

Comments

Brink showed that prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 are representable by both or neither of the quadratic forms x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2. This sequence corresponds to those representable by both, and A325080 corresponds to those representable by neither.

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 A325079

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 881:
- 881 is a prime number,
- 881 = 16*55 + 1,
- 881 = 3^2 + 3*(-8) + 14*(-8)^2 = 28^2 + 28*1 + 69*1^2,
- hence 881 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325080.

Sequence in context: A001126 A140628 A123038 * A142325 A232475 A243579

Adjacent sequences: A325076 A325077 A325078 * A325080 A325081 A325082

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved