A325067 -id:A325067 - cp's OEIS frontend

A325089 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 150*y^2.

601, 769, 2281, 2521, 2689, 3001, 5569, 5641, 5881, 6121, 6361, 6529, 6841, 7489, 8209, 8521, 9649, 9721, 11329, 12049, 12289, 12601, 13009, 14281, 14929, 15241, 16369, 17401, 17881, 18289, 19009, 19489, 19801, 20929, 21169, 21481, 21649, 21961, 22129, 22369

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 49 or 121 modulo 240 are representable by exactly one of the quadratic forms x^2 + 150*y^2 or x^2 + 960*y^2. This sequence corresponds to those representable by the first form, and A325090 corresponds to those representable by the second form.

			Regarding 5881:
- 5881 is a prime number,
- 5881 = 24*240 + 121,
- 5881 = 59^2 + 0*59*4 + 150*4^2,
- hence 5881 belongs to this sequence.

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 A325089
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325090.

PARI
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A325090 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 960*y^2.

Original entry on oeis.org

1009, 1249, 1321, 1489, 1801, 3169, 3889, 4129, 4201, 4441, 7321, 7561, 8689, 8761, 8929, 9001, 9241, 10369, 11161, 12841, 13249, 13729, 14449, 15649, 15889, 16921, 17569, 18049, 18121, 19081, 19249, 20521, 21001, 24049, 24121, 24841, 25561, 25801, 25969

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 49 or 121 modulo 240 are representable by exactly one of the quadratic forms x^2 + 150*y^2 or x^2 + 960*y^2. A325089 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

			Regarding 9001:
- 9001 is a prime number,
- 9001 = 37*240 + 121,
- 9001 = 19^2 + 960*3^2,
- hence 9001 belongs to this sequence.

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 A325090
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325089.

PARI
```
See Links section.
```

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32y^2 nor by x^2 + 64y^2.

Original entry on oeis.org

17, 97, 193, 241, 401, 433, 449, 641, 673, 769, 929, 977, 1009, 1297, 1361, 1409, 1489, 1697, 1873, 2017, 2081, 2161, 2417, 2609, 2753, 2801, 2897, 3041, 3169, 3329, 3457, 3617, 3697, 3793, 3889, 4129, 4241, 4337, 4561, 4673, 5009, 5153, 5281, 5441, 5521, 5857

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. A325067 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 17:
- 17 is a prime number,
- 17 = 16*1 + 1,
- 17 is representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2,
- hence 17 belongs to the sequence.

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 A325068
Wikipedia, Kaplansky's theorem on quadratic forms

Cf. A094407, A325067.

PARI
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See Links section.
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cp's OEIS Frontend

A325089 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 150*y^2.

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PARI

A325090 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 960*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32y^2 nor by x^2 + 64y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A325089 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 150*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325090 Prime numbers congruent to 49 or 121 modulo 240 representable by x^2 + 960*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32*y^2 nor by x^2 + 64*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325068 Prime numbers congruent to 1 modulo 16 representable neither by x^2 + 32y^2 nor by x^2 + 64y^2.