A325075 Prime numbers congruent to 1, 16 or 22 modulo 39 representable by both x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2.

139, 157, 367, 523, 547, 607, 991, 997, 1153, 1171, 1231, 1249, 1381, 1459, 1483, 1693, 1933, 1951, 2011, 2029, 2473, 2557, 3121, 3181, 3253, 3259, 3433, 3511, 3643, 3877, 4111, 4447, 4603, 4663, 4759, 5521, 5749, 5827, 6007, 6163, 6217, 6301, 6397, 6451, 6553

Offset: 1

Author

Rémy Sigrist, Mar 28 2019

Keywords

• nonn

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. This sequence corresponds to those representable by both, and A325076 corresponds to those representable by neither.

Examples

			Regarding 997:
- 997 is a prime number,
- 997 = 25*39 + 22,
- 997 = 27^2 + 27*4 + 10*4^2 = 29^2 + 29*1 + 127*1^2,
- hence 997 belongs to this sequence.
		

Links

• 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 A325075
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325076.

Programs

PARI

See Links section.