A325069 Prime numbers congruent to 9 modulo 16 representable by x^2 + 32*y^2.

41, 137, 313, 409, 457, 521, 569, 761, 809, 857, 953, 1129, 1321, 1657, 1993, 2137, 2153, 2297, 2377, 2521, 2617, 2633, 2713, 2729, 2777, 2953, 3001, 3209, 3433, 3593, 3769, 3881, 3929, 4073, 4441, 4649, 4729, 4793, 4889, 4969, 5273, 5417, 5449, 5641, 5657

Offset: 1

Author

Rémy Sigrist, Mar 27 2019

Keywords

• nonn

Comments

Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of the quadratic forms x^2 + 32*y^2 or x^2 + 64*y^2. This sequence corresponds to those representable by the first form and A325070 to those representable by the second form.

Examples

			Regarding 41:
- 41 is a prime number,
- 41 = 2*16 + 9,
- 41 = 3^2 + 32*1^2,
- hence 41 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 A325069
• Wikipedia, Kaplansky's theorem on quadratic forms

Crossrefs

See A325067 for similar results.

Cf. A105126.

Programs

PARI

See Links section.