A325076 Prime numbers congruent to 1, 16 or 22 modulo 39 neither representable by x^2 + x*y + 10*y^2 nor by x^2 + x*y + 127*y^2.

61, 79, 211, 313, 373, 601, 757, 859, 919, 937, 1069, 1093, 1303, 1327, 1543, 1621, 1699, 1777, 1873, 2083, 2089, 2161, 2239, 2341, 2551, 2707, 2713, 2731, 2791, 2887, 3019, 3331, 3571, 3727, 3823, 4057, 4273, 4423, 4507, 4657, 4813, 4969, 4993, 5209, 5227

Offset

1,1

Comments

Brink showed that prime numbers congruent to 1, 16 or 22 modulo 39 are representable by both or neither of the quadratic forms x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2. A325075 corresponds to those representable by both, and this sequence 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 A325076

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 61:
- 61 is a prime number,
- 61 = 39 + 22,
- 61 is neither representable by x^2 + x*y + 10*y^2 nor by x^2 + x*y + 127*y^2,
- hence 61 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325075.

Sequence in context: A217076 A281960 A139931 * A353598 A253232 A345239

Adjacent sequences: A325073 A325074 A325075 * A325077 A325078 A325079

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved