A325087 Prime numbers congruent to 1 or 169 modulo 240 representable by both x^2 + 150*y^2 and x^2 + 960*y^2.

1129, 3361, 3769, 4801, 5209, 5449, 5521, 5689, 8329, 8641, 9601, 9769, 10009, 10321, 10729, 12409, 13681, 15121, 15289, 15361, 15601, 16561, 16729, 17041, 17209, 17761, 18169, 18481, 20089, 21529, 21601, 23761, 24001, 24169, 25609, 25849, 26641, 26881, 27529

Offset: 1

Author

Rémy Sigrist, Mar 28 2019

Keywords

• nonn

Comments

Brink showed that prime numbers congruent to 1 or 169 modulo 240 are representable by both or neither of the quadratic forms x^2 + 150*y^2 and x^2 + 960*y^2. This sequence corresponds to those representable by both, and A325088 corresponds to those representable by neither.

Examples

			Regarding 10009:
- 10009 is a prime number,
- 10009 = 41*240 + 169,
- 10009 = 97^2 + 0*97*2 + 150*2^2 = 37^2 + 960*3^2,
- hence 10009 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 A325087
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A325088.

Programs

PARI

See Links section.