A325067 -id:A325067 - cp's OEIS frontend

A325079 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 representable by both x^2 + xy + 14y^2 and x^2 + xy + 69y^2.

71, 251, 311, 631, 661, 691, 751, 881, 1061, 1171, 1181, 1321, 1571, 1721, 1741, 1901, 1951, 2341, 2531, 2621, 2671, 2711, 2731, 2971, 3191, 3271, 3371, 3491, 3631, 3701, 3851, 3881, 4481, 4591, 4651, 5261, 5471, 5501, 5531, 5581, 5641, 5701, 5861, 6121, 6271

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 are representable by both or neither of the quadratic forms x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2. This sequence corresponds to those representable by both, and A325080 corresponds to those representable by neither.

			Regarding 881:
- 881 is a prime number,
- 881 = 16*55 + 1,
- 881 = 3^2 + 3*(-8) + 14*(-8)^2 = 28^2 + 28*1 + 69*1^2,
- hence 881 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 A325079
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325080.

PARI
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See Links section.
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A325080 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 neither representable by x^2 + xy + 14y^2 nor by x^2 + xy + 69y^2.

Original entry on oeis.org

31, 181, 191, 331, 401, 421, 521, 641, 911, 971, 991, 1021, 1291, 1301, 1511, 1621, 1831, 1871, 2011, 2161, 2281, 2311, 2381, 2861, 3001, 3041, 3061, 3221, 3301, 3331, 3391, 3821, 3931, 4051, 4211, 4261, 4271, 4621, 4691, 4801, 4871, 4931, 4951, 5021, 5171

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 are representable by both or neither of the quadratic forms x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2. A325079 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 31:
- 31 is a prime number,
- 31 = 0*55 + 31,
- 31 is neither representable by x^2 + x*y + 14*y^2 nor by x^2 + x*y + 69*y^2,
- hence 31 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 A325080
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325079.

PARI
```
See Links section.
```

A325081 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + xy + 14y^2.

Original entry on oeis.org

59, 199, 229, 269, 379, 389, 499, 509, 839, 929, 1049, 1279, 1409, 1439, 1499, 1609, 1699, 2029, 2069, 2269, 2399, 2699, 2729, 2819, 3019, 3089, 3469, 3529, 3719, 4049, 4079, 4129, 4139, 4339, 4519, 4679, 4789, 4889, 4999, 5119, 5399, 5479, 5669, 6029, 6229

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 are representable by exactly one of the quadratic forms x^2 + x*y + 14*y^2 or x^2 + x*y + 69*y^2. This sequence corresponds to those representable by the first form, and A325082 corresponds to those representable by the second form.

			Regarding 4999:
- 4999 is a prime number,
- 4999 = 90*55 + 49,
- 4999 = 41^2 + 41*14 + 14*14^2,
- hence 4999 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 A325081
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325082.

PARI
```
See Links section.
```

A325082 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + xy + 69y^2.

Original entry on oeis.org

89, 179, 419, 449, 599, 619, 709, 719, 829, 859, 1039, 1109, 1259, 1489, 1549, 1709, 1879, 2039, 2099, 2179, 2539, 2579, 2689, 2909, 3169, 3259, 3359, 3389, 3499, 3919, 4019, 4159, 4229, 4349, 4409, 4799, 4909, 5009, 5039, 5179, 5449, 5569, 5659, 5779, 5839

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 are representable by exactly one of the quadratic forms x^2 + x*y + 14*y^2 or x^2 + x*y + 69*y^2. A325081 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

			Regarding 2099:
- 2099 is a prime number,
- 2099 = 38*55 + 9,
- 2099 = 17^2 + 1*17*5 + 69*5^2,
- hence 2099 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 A325082
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325081.

PARI
```
See Links section.
```

A325083 Prime numbers congruent to 1, 65 or 81 modulo 112 representable by both x^2 + 14y^2 and x^2 + 448y^2.

Original entry on oeis.org

449, 673, 977, 1409, 1873, 2017, 2081, 2129, 2417, 2657, 2753, 3313, 3697, 4001, 4561, 4657, 4673, 4817, 4993, 6689, 6833, 7057, 7121, 7393, 7457, 7793, 8017, 8353, 8369, 8689, 8849, 9377, 9473, 9857, 10193, 10273, 11057, 11393, 11489, 11953, 12161, 12289

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 65 or 81 modulo 112 are representable by both or neither of the quadratic forms x^2 + 14*y^2 and x^2 + 448*y^2. This sequence corresponds to those representable by both, and A325084 corresponds to those representable by neither.

			Regarding 3313:
- 3313 is a prime number,
- 3313 = 29*112 + 65,
- 3313 = 53^2 + 14*6^2 = 39^2 + 448*2^2,
- hence 3313 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 A325083
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325084.

PARI
```
See Links section.
```

A325084 Prime numbers congruent to 1, 65 or 81 modulo 112 neither representable by x^2 + 14y^2 nor by x^2 + 448y^2.

Original entry on oeis.org

113, 193, 337, 401, 641, 1009, 1201, 1297, 2689, 2801, 3089, 3137, 3217, 3329, 3361, 3761, 3889, 4337, 4481, 5009, 5153, 5233, 5441, 5569, 6113, 6337, 6353, 6449, 6577, 6673, 7681, 7841, 8513, 8737, 8929, 9041, 9137, 9521, 9601, 9697, 10369, 10529, 10753

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1, 65 or 81 modulo 112 are representable by both or neither of the quadratic forms x^2 + 14*y^2 and x^2 + 448*y^2. A325083 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 113:
- 113 is a prime number,
- 113 = 1*112 + 1,
- 113 is neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2,
- hence 113 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 A325084
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325083.

PARI
```
See Links section.
```

A325085 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 14*y^2.

Original entry on oeis.org

137, 233, 281, 953, 1033, 1129, 1481, 2137, 2377, 2713, 2857, 2969, 3529, 3593, 3833, 4649, 4729, 5657, 5737, 5849, 6217, 6329, 6521, 6857, 7001, 7561, 8089, 8233, 8297, 8761, 8969, 9209, 9241, 9433, 9689, 10313, 11113, 12377, 12457, 12553, 12601, 12713, 12889

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 9, 25 or 57 modulo 112 are representable by exactly one of the quadratic forms x^2 + 14*y^2 or x^2 + 448*y^2. This sequence corresponds to those representable by the first form, and A325086 corresponds to those representable by the second form.

			Regarding 11113:
- 11113 is a prime number,
- 11113 = 99*112 + 25,
- 11113 = 103^2 + 14*6^2,
- hence 11113 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 A325085
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325086.

PARI
```
See Links section.
```

A325086 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 448*y^2.

Original entry on oeis.org

457, 569, 617, 809, 1289, 1801, 1913, 2153, 2297, 2473, 2521, 2633, 3049, 3257, 3929, 4057, 4153, 4201, 4937, 5209, 5273, 5881, 6073, 6553, 6841, 7177, 7193, 7417, 7529, 7673, 7753, 8009, 8521, 8537, 8681, 9769, 10889, 11257, 11321, 11369, 11593, 11657, 11897

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 9, 25 or 57 modulo 112 are representable by exactly one of the quadratic forms x^2 + 14*y^2 or x^2 + 448*y^2. A325085 corresponds to those representable by the first form, and this sequence corresponds to those representable by the second form.

			Regarding 7177:
- 7177 is a prime number,
- 7177 = 64*112 + 9,
- 7177 = 3^2 + 448*4^2,
- hence 7177 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 A325086
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325085.

PARI
```
See Links section.
```

A325087 Prime numbers congruent to 1 or 169 modulo 240 representable by both x^2 + 150y^2 and x^2 + 960y^2.

Original entry on oeis.org

1129, 3361, 3769, 4801, 5209, 5449, 5521, 5689, 8329, 8641, 9601, 9769, 10009, 10321, 10729, 12409, 13681, 15121, 15289, 15361, 15601, 16561, 16729, 17041, 17209, 17761, 18169, 18481, 20089, 21529, 21601, 23761, 24001, 24169, 25609, 25849, 26641, 26881, 27529

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1 or 169 modulo 240 are representable by both or neither of the quadratic forms x^2 + 150*y^2 and x^2 + 960*y^2. This sequence corresponds to those representable by both, and A325088 corresponds to those representable by neither.

			Regarding 10009:
- 10009 is a prime number,
- 10009 = 41*240 + 169,
- 10009 = 97^2 + 0*97*2 + 150*2^2 = 37^2 + 960*3^2,
- hence 10009 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 A325087
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325088.

PARI
```
See Links section.
```

A325088 Prime numbers congruent to 1 or 169 modulo 240 representable neither by x^2 + 150y^2 nor by x^2 + 960y^2.

Original entry on oeis.org

241, 409, 1201, 1609, 2089, 2161, 3049, 3121, 3529, 4561, 4729, 4969, 5281, 6481, 6961, 7129, 7369, 7681, 8089, 8161, 9049, 11689, 12241, 12721, 12889, 13441, 13921, 14401, 16249, 17449, 17929, 19441, 19609, 19681, 20161, 20641, 20809, 21121, 21841, 23041

Offset: 1

Rémy Sigrist, Mar 28 2019

nonn

Brink showed that prime numbers congruent to 1 or 169 modulo 240 are representable by both or neither of the quadratic forms x^2 + 150*y^2 and x^2 + 960*y^2. A325087 corresponds to those representable by both, and this sequence corresponds to those representable by neither.

			Regarding 241:
- 241 is a prime number,
- 241 = 1*240 + 1,
- 241 is neither representable by x^2 + 150*y^2 nor by x^2 + 960*y^2,
- hence 241 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 A325088
Wikipedia, Kaplansky's theorem on quadratic forms

See A325067 for similar results.

Cf. A325087.

PARI
```
See Links section.
```

cp's OEIS Frontend

A325079 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 representable by both x^2 + x*y + 14*y^2 and x^2 + x*y + 69*y^2.

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A325080 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 neither representable by x^2 + x*y + 14*y^2 nor by x^2 + x*y + 69*y^2.

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PARI

A325081 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + x*y + 14*y^2.

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PARI

A325082 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + x*y + 69*y^2.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325083 Prime numbers congruent to 1, 65 or 81 modulo 112 representable by both x^2 + 14*y^2 and x^2 + 448*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325084 Prime numbers congruent to 1, 65 or 81 modulo 112 neither representable by x^2 + 14*y^2 nor by x^2 + 448*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325085 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 14*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A325086 Prime numbers congruent to 9, 25 or 57 modulo 112 representable by x^2 + 448*y^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A325079 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 representable by both x^2 + xy + 14y^2 and x^2 + xy + 69y^2.

A325080 Prime numbers congruent to 1, 16, 26, 31 or 36 modulo 55 neither representable by x^2 + xy + 14y^2 nor by x^2 + xy + 69y^2.

A325081 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + xy + 14y^2.

A325082 Prime numbers congruent to 4, 9, 14, 34 or 49 modulo 55 representable by x^2 + xy + 69y^2.

A325083 Prime numbers congruent to 1, 65 or 81 modulo 112 representable by both x^2 + 14y^2 and x^2 + 448y^2.

A325084 Prime numbers congruent to 1, 65 or 81 modulo 112 neither representable by x^2 + 14y^2 nor by x^2 + 448y^2.

A325087 Prime numbers congruent to 1 or 169 modulo 240 representable by both x^2 + 150y^2 and x^2 + 960y^2.

A325088 Prime numbers congruent to 1 or 169 modulo 240 representable neither by x^2 + 150y^2 nor by x^2 + 960y^2.