A102275 -id:A102275 - cp's OEIS frontend

A102273 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = +1.

11, 23, 71, 107, 179, 191, 239, 263, 347, 359, 431, 443, 491, 599, 659, 683, 743, 827, 863, 911, 947, 1019, 1031, 1103, 1163, 1187, 1283, 1367, 1439, 1451, 1499, 1523, 1583, 1607, 1619, 1667, 1787, 1871, 2003, 2027, 2039, 2087

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

The 2-class number of these fields is always 4.
Primes of the form 2x^2 - 2xy + 11y^2 with x nonnegative and y positive. - T. D. Noe, May 08 2005
Also primes of the forms 8x^2 + 4xy + 11y^2 and 11x^2 + 2xy + 23y^2. See A140633. - T. D. Noe, May 19 2008
The discriminant of positive definite binary quadratic form (2,2,11) is -84. - Hugo Pfoertner, Jul 14 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A102269-A102275.

Cf. A139827.

[p: p in PrimesUpTo(3000) | p mod 84 in [2, 11, 23, 71]]; // Vincenzo Librandi, Jul 19 2012

f[x_,y_]:=2*x^2+2*x*y+11*y^2; lst={};Do[Do[p=f[x,y];If[PrimeQ[p],AppendTo[lst,p]],{y,-5!,6!}],{x,-5!,6!}];Take[Union[lst],5! ] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2009 *)

The primes are congruent to {2, 11, 23, 71} (mod 84). - T. D. Noe, May 02 2008

A102271 Primes of the form 3x^2 + 7y^2.

Original entry on oeis.org

3, 7, 19, 31, 103, 139, 199, 223, 271, 283, 307, 367, 439, 523, 607, 619, 643, 691, 727, 787, 811, 859, 1039, 1063, 1123, 1231, 1279, 1291, 1399, 1447, 1459, 1483, 1531, 1543, 1567, 1627, 1699, 1783, 1867, 1879, 1951, 1987, 2131, 2203, 2239, 2287, 2371, 2383

Offset: 1

N. J. A. Sloane, Feb 19 2005

Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = -1.

Ray Chandler, Table of n, a(n) for n = 1..10000 (First 1000 terms from Vincenzo Librandi).
H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A102269-A102275, A139827.

[p: p in PrimesUpTo(3000) | p mod 84 in [3, 7, 19, 31, 55]]; // Vincenzo Librandi, Jul 19 2012

m=3; n=7; pLst={}; lim=3000; xMax=Sqrt[lim/m]; yMax=Sqrt[lim/n]; Do[p=m*x^2+n*y^2; If[pT. D. Noe, May 05 2005 *)
QuadPrimes2[3, 0, 7, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\7), if(isprime(t=w+7*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {3, 7, 19, 31, 55} (mod 84). - T. D. Noe, May 02 2008

A102269 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = +1.

Original entry on oeis.org

43, 67, 79, 127, 151, 163, 211, 331, 379, 463, 487, 499, 547, 571, 631, 739, 751, 823, 883, 907, 919, 967, 991, 1051, 1087, 1171, 1303, 1327, 1423, 1471, 1579, 1663, 1723, 1747, 1759, 1831, 1999, 2011, 2083, 2143, 2179, 2251, 2311, 2347, 2503, 2647, 2671, 2683, 2731, 2767, 2851, 3019

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

Primes p such that p is 3 (mod 4) and (-3/p) = (-7/p) = 1, where (k/n) is the Kronecker symbol. - Robin Visser, Mar 13 2024

H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.

Cf. A102270, A102271, A102272, A102273, A102274, A102275.

[p : p in PrimesUpTo(3000) | p mod 84 in [43, 67, 79]];  // Robin Visser, Mar 13 2024

The primes are congruent to {43, 67, 79} (mod 84). - Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

A102270 2-class number of Q(sqrt(-21p)) as p runs through primes in A102269.

Original entry on oeis.org

16, 8, 8, 8, 16, 8, 16, 16, 16, 8, 16, 32, 8, 8, 8, 8, 32, 8, 16, 64, 32, 8, 32, 8, 8, 128, 32, 32, 32, 16, 8, 8, 8, 16, 16, 8, 8, 8, 128, 16, 8, 8, 16, 32, 8, 16, 32, 8, 8, 8, 128, 8, 8, 8, 16, 8, 64, 8, 8, 8, 16, 8, 16, 16, 16, 8, 8, 16, 32, 16, 8, 8, 8, 64, 32, 64, 16, 64, 8, 16, 32

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.

Cf. A102269, A102271, A102272, A102273, A102274, A102275.

[2^QuadraticField(-21*p).class_number().valuation(2) for p in Primes()[:1000] if (p%84) in [43, 67, 79]] # Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

A102272 2-class number of Q(sqrt(-21p)) as p runs through primes in A102271.

Original entry on oeis.org

16, 8, 8, 8, 16, 16, 8, 32, 16, 16, 16, 8, 16, 8, 64, 8, 8, 8, 8, 128, 16, 8, 8, 32

Offset: 1

N. J. A. Sloane, Feb 19 2005

H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.

Cf. A102269, A102270, A102271, A102273, A102274, A102275.

A102274 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = -1.

Original entry on oeis.org

47, 59, 83, 131, 167, 227, 251, 311, 383, 419, 467, 479, 503, 563, 587, 647, 719, 839, 887, 971, 983, 1091, 1151, 1223, 1259, 1307, 1319, 1427, 1487, 1511, 1559, 1571, 1811, 1823, 1847, 1907, 1931, 1979, 2063, 2099, 2243, 2267, 2351, 2399, 2411, 2579, 2663, 2687, 2819, 2903, 2939, 2999

Offset: 1

N. J. A. Sloane Feb 19 2005

nonn

Primes p such that p is 3 (mod 4) and (-3/p) = (-7/p) = -1, where (k/n) is the Kronecker symbol. - Robin Visser, Mar 13 2024

H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.

Cf. A102269, A102270, A102271, A102272, A102273, A102275.

[p : p in PrimesUpTo(3000) | p mod 84 in [47, 59, 83]];  // Robin Visser, Mar 13 2024

The primes are congruent to {47, 59, 83} (mod 84). - Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

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A102273 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = +1.

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A102271 Primes of the form 3*x^2 + 7*y^2.

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A102269 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = +1.

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A102270 2-class number of Q(sqrt(-21p)) as p runs through primes in A102269.

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A102272 2-class number of Q(sqrt(-21p)) as p runs through primes in A102271.

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A102274 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = -1.

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Magma

Formula

Extensions

A102271 Primes of the form 3x^2 + 7y^2.