A035120 -id:A035120 - cp's OEIS frontend

A003656 Discriminants of real quadratic fields with unique factorization.

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269

Offset: 1

N. J. A. Sloane, Mira Bernstein

Discriminants of real quadratic fields with class number 1.

Other than the term 8, every term is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. In fact, for a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of the form (i), (ii) or (iii). See Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003652, A003658, A014602 (imaginary case).

For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

maxDisc = 269; t = Table[ {NumberFieldDiscriminant[ Sqrt[n] ], NumberFieldClassNumber[ Sqrt[n] ]}, {n, Select[ Range[2, maxDisc], SquareFreeQ] } ]; Union[ Select[ t, #[[2]] == 1 && #[[1]] <= maxDisc & ][[All, 1]]] (* Jean-François Alcover, Jan 24 2012 *)

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
A003656(270) # Peter Luschny, Aug 10 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A052475 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(5)).

Original entry on oeis.org

40, 60, 65, 85, 105, 120, 140, 165, 185, 205, 220, 265, 280, 285, 305, 345, 365, 380, 385, 440, 460, 465, 485, 545, 565, 620, 645, 665, 685, 705, 745, 760, 805, 860, 865, 885, 920, 965, 1005, 1065, 1085, 1165, 1180, 1185, 1205, 1240, 1245, 1265, 1285

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.

Cf. A035120, A003656, A052476, A052477, A052478, A052479.

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A052476 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(2)).

Original entry on oeis.org

104, 136, 168, 232, 264, 296, 424, 456, 488, 552, 584, 616, 712, 744, 776, 808, 872, 1032, 1064, 1128, 1192, 1256, 1416, 1448, 1544, 1576, 1608, 1672, 1704, 1832, 1864, 1896, 1928, 1992

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.

Cf. A035120, A003656, A052475-...

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A052477 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(3)).

Original entry on oeis.org

156, 204, 348, 444, 492, 636, 732, 1068, 1212, 1308, 1356, 1644, 1788, 1884

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Cf. A035120, A003656, A052475-...

A052478 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(13)).

Original entry on oeis.org

221, 273, 312, 364, 377, 429, 481, 533, 572, 728, 741, 949, 988, 1001, 1144, 1157, 1196, 1209, 1261, 1417, 1469, 1612, 1729, 1781, 1833, 1976

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Cf. A035120, A003656, A052475-...

A052479 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(17)).

Original entry on oeis.org

357, 408, 476, 493, 561, 629, 748, 952, 969, 1037, 1173, 1241, 1309, 1496, 1513, 1564, 1581, 1649, 1717, 1853, 1921

Offset: 1

N. J. A. Sloane, Mar 15 2000

H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, pp. 534-535.

Cf. A035120, A003656, A052475-...

cp's OEIS Frontend

A003656 Discriminants of real quadratic fields with unique factorization.

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A052475 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(5)).

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A052476 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(2)).

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A052477 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(3)).

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A052478 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(13)).

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A052479 Discriminants of real quadratic number fields with class number 2 such that Hilbert class field has splitting field Q(sqrt(17)).

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