A325074 -id:A325074 - cp's OEIS frontend

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32y^2 and x^2 + 64y^2.

113, 257, 337, 353, 577, 593, 881, 1153, 1201, 1217, 1249, 1553, 1601, 1777, 1889, 2113, 2129, 2273, 2593, 2657, 2689, 2833, 3089, 3121, 3137, 3217, 3313, 3361, 3761, 4001, 4049, 4177, 4273, 4289, 4481, 4513, 4657, 4721, 4801, 4817, 4993, 5233, 5297, 5393

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. This sequence corresponds to those representable by both, and A325068 corresponds to those representable by neither.
Also, Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of these quadratic forms. A325069 corresponds to those representable by the first form and A325070 to those representable by the second form.
Brink provided similar results for other congruences.

			Regarding 1201:
- 1201 is a prime number,
- 1201 = 75*16 + 1,
- 1201 = 7^2 + 32*6^2 = 25^2 + 64*3^2,
- hence 1201 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 A325067
Wikipedia, Kaplansky's theorem on quadratic forms

Cf. A094407, A105126, A325068, A325069, A325070.
See A325071, A325072, A325073 and A325074 for similar results in congruences modulo 16.
See A325075, A325076, A325077 and A325078 for similar results in congruences modulo 39.
See A325079, A325080, A325081 and A325082 for similar results in congruences modulo 55.
See A325083, A325084, A325085 and A325086 for similar results in congruences modulo 112.
See A325087, A325088, A325089 and A325090 for similar results in congruences modulo 240.

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A325073 Prime numbers congruent to 9 modulo 20 representable by x^2 + 20*y^2.

Original entry on oeis.org

29, 89, 229, 349, 509, 709, 769, 809, 1009, 1049, 1109, 1229, 1249, 1289, 1409, 1549, 1669, 1709, 1789, 2029, 2069, 2089, 2389, 2729, 3049, 3089, 3169, 3329, 3389, 3469, 3529, 3929, 3989, 4049, 4229, 4289, 4549, 4649, 4729, 4789, 5009, 5209, 5669, 5689, 5849

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Brink showed that prime numbers congruent to 9 modulo 20 are representable by exactly one of the quadratic forms x^2 + 20*y^2 or x^2 + 100*y^2. This sequence corresponds to those representable by the first form, and A325074 corresponds to those representable by the second form.

			Regarding 1009:
- 1009 is a prime number,
- 1009 = 50*20 + 9,
- 1009 = 17^2 + 20*6^2,
- hence 1009 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 A325073
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A141883, A325074.

PARI
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See Links section.
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cp's OEIS Frontend

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32y^2 and x^2 + 64y^2.

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A325073 Prime numbers congruent to 9 modulo 20 representable by x^2 + 20*y^2.

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cp's OEIS Frontend

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32*y^2 and x^2 + 64*y^2.

Views

Author

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PARI

A325073 Prime numbers congruent to 9 modulo 20 representable by x^2 + 20*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32y^2 and x^2 + 64y^2.