A325071 Prime numbers congruent to 1 modulo 20 representable by both x^2 + 20*y^2 and x^2 + 100*y^2.

101, 181, 401, 461, 521, 541, 761, 941, 1021, 1061, 1361, 1601, 1621, 1721, 1741, 1861, 2081, 2441, 2621, 2801, 2861, 3001, 3121, 3301, 3461, 3581, 3821, 3881, 4001, 4021, 4201, 4441, 4561, 4621, 4861, 5021, 5081, 5101, 5261, 5281, 5441, 5741, 5861, 5981, 6221

Offset: 1

Author

Rémy Sigrist, Mar 27 2019

Keywords

• nonn

Comments

Brink showed that prime numbers congruent to 1 modulo 20 are representable by both or neither of the quadratic forms x^2 + 20*y^2 and x^2 + 100*y^2. This sequence corresponds to those representable by both, and A325072 corresponds to those representable by neither.

Examples

			Regarding 1601:
- 1601 is a prime number,
- 1601 = 80*20 + 1,
- 1601 = 39^2 + 20*2^2 = 1^2 + 100*4^2,
- hence 1601 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 A325071
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A141881, A325072.

Programs

PARI

See Links section.