A325077 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + x*y + 10*y^2.

43, 103, 181, 277, 439, 673, 751, 823, 1039, 1063, 1117, 1429, 1453, 1759, 1993, 1999, 2131, 2287, 2311, 2467, 2521, 2539, 2617, 2833, 2851, 2857, 3067, 3163, 3457, 3559, 3613, 3637, 3847, 3943, 4021, 4027, 4177, 4261, 4339, 4723, 4783, 4861, 5113, 5119, 5197

Offset

1,1

Comments

Brink showed that prime numbers congruent to 4, 10 or 25 modulo 39 are representable by exactly one of the quadratic forms x^2 + x*y + 10*y^2 or x^2 + x*y + 127*y^2. This sequence corresponds to those representable by the first form, and A325078 corresponds to those representable by the second form.

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 A325077

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 43:
- 43 is a prime number,
- 43 = 39 + 4,
- 43 = 1^2 + 1*2 + 10*2^2,
- hence 43 belongs to this sequence.

Prog

(PARI) See Links section.

Crossrefs

See A325067 for similar results.

Cf. A325078.

Sequence in context: A033227 A106888 A142795 * A023293 A115606 A194773

Adjacent sequences: A325074 A325075 A325076 * A325078 A325079 A325080

Keyword

nonn

Author

Rémy Sigrist, Mar 28 2019

Status

approved