A325072 Prime numbers congruent to 1 modulo 20 neither representable by x^2 + 20*y^2 nor by x^2 + 100*y^2.

41, 61, 241, 281, 421, 601, 641, 661, 701, 821, 881, 1181, 1201, 1301, 1321, 1381, 1481, 1801, 1901, 2141, 2161, 2221, 2281, 2341, 2381, 2521, 2741, 3041, 3061, 3181, 3221, 3361, 3541, 3701, 3761, 4241, 4261, 4421, 4481, 4721, 4801, 5381, 5501, 5521, 5581

Offset

1,1

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. A325071 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

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.

Akihide Hanaki, Kenji Kobayashi, and Akihiro Munemasa, 3-Designs from PSL(2,q) with cyclic starter blocks, arXiv:2502.13331 [math.CO], 2025. See pp. 2, 9. See also J. Combin. Des. (2025).

Rémy Sigrist, PARI program for A325072

Wikipedia, Kaplansky's theorem on quadratic forms

Example

Regarding 2221:
- 2221 is a prime number,
- 2221 = 111*20 + 1,
- 2221 is neither representable by x^2 + 20*y^2 nor by x^2 + 100*y^2,
- hence 2221 belongs to this sequence.

Prog

(PARI) \\ See Links section.

Crossrefs

See A325067 for similar results.

Cf. A141881, A325071.

Sequence in context: A260554 A384490 A031415 * A089345 A320468 A055110

Adjacent sequences: A325069 A325070 A325071 * A325073 A325074 A325075

Keyword

nonn

Author

Rémy Sigrist, Mar 27 2019

Status

approved