A325067 -id:A325067 - cp's OEIS frontend

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

Rémy Sigrist, Mar 27 2019

nonn

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.

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

See A325067 for similar results.

Cf. A105126.

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See Links section.
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A325070 Prime numbers congruent to 9 modulo 16 representable by x^2 + 64*y^2.

Original entry on oeis.org

73, 89, 233, 281, 601, 617, 937, 1033, 1049, 1097, 1193, 1289, 1433, 1481, 1609, 1721, 1753, 1801, 1913, 2089, 2281, 2393, 2441, 2473, 2857, 2969, 3049, 3257, 3449, 3529, 3673, 3833, 4057, 4153, 4201, 4217, 4297, 4409, 4457, 4937, 5081, 5113, 5209, 5689, 5737

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

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. A325069 corresponds to those representable by the first form and this sequence to those representable by the second form.

			Regarding 4201:
- 4201 is a prime number,
- 4201 = 262*16 + 9,
- 4201 = 51^2 + 64*5^2,
- hence 4201 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 A325070
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A105126, A325069.

PARI
```
See Links section.
```

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

Original entry on oeis.org

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

Rémy Sigrist, Mar 27 2019

nonn

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.

			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.

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

See A325067 for similar results.

Cf. A141881, A325072.

PARI
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See Links section.
```

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

Original entry on oeis.org

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

Rémy Sigrist, Mar 27 2019

nonn

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.

			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.

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, 13.
Rémy Sigrist, PARI program for A325072
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A141881, A325071.

PARI
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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
```
See Links section.
```

A325074 Prime numbers congruent to 9 modulo 20 representable by x^2 + 100*y^2.

Original entry on oeis.org

109, 149, 269, 389, 409, 449, 569, 829, 929, 1069, 1129, 1429, 1489, 1609, 1889, 1949, 2129, 2269, 2309, 2549, 2609, 2689, 2749, 2789, 2909, 2969, 3109, 3209, 3229, 3449, 3709, 3769, 3889, 4129, 4349, 4409, 4889, 4909, 4969, 5189, 5309, 5449, 5569, 5749, 6029

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. A325073 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

			Regarding 4409:
- 4409 is a prime number,
- 4409 = 220*20 + 9,
- 4409 = 53^2 + 100*4^2,
- hence 4409 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 A325074
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A141883, A325073.

PARI
```
See Links section.
```

A325075 Prime numbers congruent to 1, 16 or 22 modulo 39 representable by both x^2 + xy + 10y^2 and x^2 + xy + 127y^2.

Original entry on oeis.org

139, 157, 367, 523, 547, 607, 991, 997, 1153, 1171, 1231, 1249, 1381, 1459, 1483, 1693, 1933, 1951, 2011, 2029, 2473, 2557, 3121, 3181, 3253, 3259, 3433, 3511, 3643, 3877, 4111, 4447, 4603, 4663, 4759, 5521, 5749, 5827, 6007, 6163, 6217, 6301, 6397, 6451, 6553

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 16 or 22 modulo 39 are representable by both or neither of the quadratic forms x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2. This sequence corresponds to those representable by both, and A325076 corresponds to those representable by neither.

			Regarding 997:
- 997 is a prime number,
- 997 = 25*39 + 22,
- 997 = 27^2 + 27*4 + 10*4^2 = 29^2 + 29*1 + 127*1^2,
- hence 997 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 A325075
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325076.

PARI
```
See Links section.
```

A325076 Prime numbers congruent to 1, 16 or 22 modulo 39 neither representable by x^2 + xy + 10y^2 nor by x^2 + xy + 127y^2.

Original entry on oeis.org

61, 79, 211, 313, 373, 601, 757, 859, 919, 937, 1069, 1093, 1303, 1327, 1543, 1621, 1699, 1777, 1873, 2083, 2089, 2161, 2239, 2341, 2551, 2707, 2713, 2731, 2791, 2887, 3019, 3331, 3571, 3727, 3823, 4057, 4273, 4423, 4507, 4657, 4813, 4969, 4993, 5209, 5227

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 16 or 22 modulo 39 are representable by both or neither of the quadratic forms x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2. A325075 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 61:
- 61 is a prime number,
- 61 = 39 + 22,
- 61 is neither representable by x^2 + x*y + 10*y^2 nor by x^2 + x*y + 127*y^2,
- hence 61 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 A325076
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325075.

PARI
```
See Links section.
```

A325077 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + xy + 10y^2.

Original entry on oeis.org

43, 103, 181, 277, 439, 673, 751, 823, 1039, 1063, 1117, 1429, 1453, 1759, 1993, 1999, 2131, 2287, 2311, 2467, 2521, 2539, 2617, 2833, 2851, 2857, 3067, 3163, 3457, 3559, 3613, 3637, 3847, 3943, 4021, 4027, 4177, 4261, 4339, 4723, 4783, 4861, 5113, 5119, 5197

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 4, 10 or 25 modulo 39 are representable by exactly one of the quadratic forms x^2 + x*y + 10*y^2 or x^2 + x*y + 127*y^2. This sequence corresponds to those representable by the first form, and A325078 corresponds to those representable by the second form.

			Regarding 43:
- 43 is a prime number,
- 43 = 39 + 4,
- 43 = 1^2 + 1*2 + 10*2^2,
- hence 43 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 A325077
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325078.

PARI
```
See Links section.
```

A325078 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + xy + 127y^2.

Original entry on oeis.org

127, 199, 283, 337, 433, 571, 727, 829, 883, 907, 1213, 1291, 1297, 1447, 1531, 1609, 1663, 1741, 2053, 2383, 2389, 2677, 3169, 3301, 3319, 3631, 3691, 3709, 3769, 3793, 4003, 4099, 4159, 4549, 4567, 4651, 4729, 4801, 4957, 5347, 5407, 5431, 5563, 5821, 6133

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 4, 10 or 25 modulo 39 are representable by exactly one of the quadratic forms x^2 + x*y + 10*y^2 or x^2 + x*y + 127*y^2. A325077 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

			Regarding 127:
- 127 is a prime number,
- 127 = 3*39 + 10,
- 127 = 0^2 + 0*1 + 127*1^2,
- hence 127 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 A325078
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325077.

PARI
```
See Links section.
```

cp's OEIS Frontend

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

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A325070 Prime numbers congruent to 9 modulo 16 representable by x^2 + 64*y^2.

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

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PARI

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

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PARI

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325074 Prime numbers congruent to 9 modulo 20 representable by x^2 + 100*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325075 Prime numbers congruent to 1, 16 or 22 modulo 39 representable by both x^2 + x*y + 10*y^2 and x^2 + x*y + 127*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325076 Prime numbers congruent to 1, 16 or 22 modulo 39 neither representable by x^2 + x*y + 10*y^2 nor by x^2 + x*y + 127*y^2.

Views

Author

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Comments

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

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

A325075 Prime numbers congruent to 1, 16 or 22 modulo 39 representable by both x^2 + xy + 10y^2 and x^2 + xy + 127y^2.

A325076 Prime numbers congruent to 1, 16 or 22 modulo 39 neither representable by x^2 + xy + 10y^2 nor by x^2 + xy + 127y^2.

A325077 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + xy + 10y^2.

A325078 Prime numbers congruent to 4, 10 or 25 modulo 39 representable by x^2 + xy + 127y^2.