## Class Number

For any Ideal , there is an Ideal such that

 (1)

where is a Principal Ideal, (i.e., an Ideal of rank 1). Moreover, there is a finite list of ideals such that this equation may be satisfied for every . The size of this list is known as the class number. When the class number is 1, the Ring corresponding to a given Ideal has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in the original number ring.

A finite series giving exactly the class number of a Ring is known as a Class Number Formula. A Class Number Formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. Finding the class number is a computationally difficult problem.

Let denote the class number of a quadratic ring, corresponding to the Binary Quadratic Form

 (2)

with Discriminant
 (3)

Then the class number for Discriminant gives the number of possible factorizations of in the Quadratic Field . Here, the factors are of the form , with and half Integers.

Some fairly sophisticated mathematics shows that the class number for discriminant can be given by the Class Number Formula

 (4)

where is the Kronecker Symbol, is the Fundamental Unit, is the number of substitutions which leave the Binary Quadratic Form unchanged
 (5)

and the sums are taken over all terms where the Kronecker Symbol is defined (Cohn 1980). The class number for can also be written
 (6)

for , where the Product is taken over terms for which the Kronecker Symbol is defined.

The class number is related to the Dirichlet L-Series by

 (7)

where is the Dirichlet Structure Constant.

Wagner (1996) shows that class number satisfies the Inequality

 (8)

for , where is the Floor Function, the product is over Primes dividing , and the indicates that the Greatest Prime Factor of is omitted from the product.

The Mathematica (Wolfram Research, Champaign, IL) function NumberTheoryNumberTheoryFunctionsClassNumber[n] gives the class number for a Negative Squarefree number of the form .

Gauss's Class Number Problem asks to determine a complete list of fundamental Discriminants such that the Class Number is given by for a given . This problem has been solved for and Odd . Gauß conjectured that the class number of an Imaginary quadratic field with Discriminant tends to infinity with , an assertion now known as Gauss's Class Number Conjecture.

The discriminants having , 2, 3, 4, 5, ... are Sloane's A014602 (Cohen 1993, p. 229; Cox 1997, p. 271), Sloane's A014603 (Cohen 1993, p. 229), Sloane's A006203 (Cohen 1993, p. 504), Sloane's A013658 (Cohen 1993, p. 229), Sloane's A046002, Sloane's A046003, .... The complete set of negative discriminants having class numbers 1-5 and Odd 7-23 are known. Buell (1977) gives the smallest and largest fundamental class numbers for , partitioned into Even discriminants, discriminants 1 (mod 8), and discriminants 5 (mod 8). Arno et al. (1993) give complete lists of values of with for Odd , 7, 9, ..., 23. Wagner gives complete lists of values for , 6, and 7.

Lists of Negative discriminants corresponding to Imaginary Quadratic Fields having small class numbers are given in the table below. In the table, is the number of fundamental'' values of with a given class number , where fundamental'' means that is not divisible by any Square Number such that . For example, although , is not a fundamental discriminant since and . Even values have been computed by Weisstein. The number of negative discriminants having class number 1, 2, 3, ... are 9, 18, 16, 54, 25, 51, 31, ... (Sloane's A046125). The largest negative discriminants having class numbers 1, 2, 3, ... are 163, 427, 907, 1555, 2683, ... (Sloane's A038552).

The following table lists the numbers having class numbers . The search was terminated at 50000, 70000, 90000, and 90000 for class numbers 18, 20, 22, and 24, respectively. As far as I know, analytic upper bounds are not currently known for these cases.

 Sloane 1 9 Sloane's A014602 3, 4, 7, 8, 11, 19, 43, 67, 163 2 18 Sloane's A014603 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427 3 16 Sloane's A006203 23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907 4 54 Sloane's A013658 39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555 5 25 Sloane's A046002 47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947, 1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683 6 51 Sloane's A046003 87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835, 843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763 7 31 Sloane's A046004 71, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787, 1987, 2011, 2083, 2179, 2251, 2467, 2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923 8 131 Sloane's A046005 95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651, 660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240, 1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528, 1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780, 1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995, 2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308, 2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715, 2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315, 3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195, 4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707, 5947, 6307 9 34 Sloane's A046006 199, 367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 2003, 2803, 3163, 3259, 3307, 3547, 3643, 4027, 4243, 4363, 4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563, 8803, 9067, 10627 10 87 Sloane's A046007 119, 143, 159, 296, 303, 319, 344, 415, 488, 611, 635, 664, 699, 724, 779, 788, 803, 851, 872, 916, 923, 1115, 1268, 1384, 1492, 1576, 1643, 1684, 1688, 1707, 1779, 1819, 1835, 1891, 1923, 2152, 2164, 2363, 2452, 2643, 2776, 2836, 2899, 3028, 3091, 3139, 3147, 3291, 3412, 3508, 3635, 3667, 3683, 3811, 3859, 3928, 4083, 4227, 4372, 4435, 4579, 4627, 4852, 4915, 5131, 5163, 5272, 5515, 5611, 5667, 5803, 6115, 6259, 6403, 6667, 7123, 7363, 7387, 7435, 7483, 7627, 8227, 8947, 9307, 10147, 10483, 13843 11 41 Sloane's A046008 167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203, 3347, 3499, 3739, 3931, 4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443, 9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683, 15667 12 206 Sloane's A046009 231, 255, 327, 356, 440, 516, 543, 655, 680, 687, 696, 728, 731, 744, 755, 804, 888, 932, 948, 964, 984, 996, 1011, 1067, 1096, 1144, 1208, 1235, 1236, 1255, 1272, 1336, 1355, 1371, 1419, 1464, 1480, 1491, 1515, 1547, 1572, 1668, 1720, 1732, 1763, 1807, 1812, 1892, 1955, 1972, 2068, 2091, 2104, 2132, 2148, 2155, 2235, 2260, 2355, 2387, 2388, 2424, 2440, 2468, 2472, 2488, 2491, 2555, 2595, 2627, 2635, 2676, 2680, 2692, 2723, 2728, 2740, 2795, 2867, 2872, 2920, 2955, 3012, 3027, 3043, 3048, 3115, 3208, 3252, 3256, 3268, 3304, 3387, 3451, 3459, 3592, 3619, 3652, 3723, 3747, 3768, 3796, 3835, 3880, 3892, 3955, 3972, 4035, 4120, 4132, 4147, 4152, 4155, 4168, 4291, 4360, 4411, 4467, 4531, 4552, 4555, 4587, 4648, 4699, 4708, 4755, 4771, 4792, 4795, 4827, 4888, 4907, 4947, 4963, 5032, 5035, 5128, 5140, 5155, 5188, 5259, 5299, 5307, 5371, 5395, 5523, 5595, 5755, 5763, 5811, 5835, 6187, 6232, 6235, 6267, 6283, 6472, 6483, 6603, 6643, 6715, 6787, 6843, 6931, 6955, 6963, 6987, 7107, 7291, 7492, 7555, 7683, 7891, 7912, 8068, 8131, 8155, 8248, 8323, 8347, 8395, 8787, 8827, 9003, 9139, 9355, 9523, 9667, 9843, 10003, 10603, 10707, 10747, 10795, 10915, 11155, 11347, 11707, 11803, 12307, 12643, 14443, 15163, 15283, 16003, 17803 13 37 Sloane's A046010 191, 263, 607, 631, 727, 1019, 1451, 1499, 1667, 1907, 2131, 2143, 2371, 2659, 2963, 3083, 3691, 4003, 4507, 4643, 5347, 5419, 5779, 6619, 7243, 7963, 9547, 9739, 11467, 11587, 11827, 11923, 12043, 14347, 15787, 16963, 20563 14 96 Sloane's A046011 215, 287, 391, 404, 447, 511, 535, 536, 596, 692, 703, 807, 899, 1112, 1211, 1396, 1403, 1527, 1816, 1851, 1883, 2008, 2123, 2147, 2171, 2335, 2427, 2507, 2536, 2571, 2612, 2779, 2931, 2932, 3112, 3227, 3352, 3579, 3707, 3715, 3867, 3988, 4187, 4315, 4443, 4468, 4659, 4803, 4948, 5027, 5091, 5251, 5267, 5608, 5723, 5812, 5971, 6388, 6499, 6523, 6568, 6979, 7067, 7099, 7147, 7915, 8035, 8187, 8611, 8899, 9115, 9172, 9235, 9427, 10123, 10315, 10363, 10411, 11227, 12147, 12667, 12787, 13027, 13435, 13483, 13603, 14203, 16867, 18187, 18547, 18643, 20227, 21547, 23083, 23692, 30067 15 68 Sloane's A046012 239, 439, 751, 971, 1259, 1327, 1427, 1567, 1619, 2243, 2647, 2699, 2843, 3331, 3571, 3803, 4099, 4219, 5003, 5227, 5323, 5563, 5827, 5987, 6067, 6091, 6211, 6571, 7219, 7459, 7547, 8467, 8707, 8779, 9043, 9907, 10243, 10267, 10459, 10651, 10723, 11083, 11971, 12163, 12763, 13147, 13963, 14323, 14827, 14851, 15187, 15643, 15907, 16603, 16843, 17467, 17923, 18043, 18523, 19387, 19867, 20707, 22003, 26203, 27883, 29947, 32323, 34483 16 322 Sloane's A046013 399, 407, 471, 559, 584, 644, 663, 740, 799, 884, 895, 903, 943, 1015, 1016, 1023, 1028, 1047, 1139, 1140, 1159, 1220, 1379, 1412, 1416, 1508, 1560, 1595, 1608, 1624, 1636, 1640, 1716, 1860, 1876, 1924, 1983, 2004, 2019, 2040, 2056, 2072, 2095, 2195, 2211, 2244, 2280, 2292, 2296, 2328, 2356, 2379, 2436, 2568, 2580, 2584, 2739, 2760, 2811, 2868, 2884, 2980, 3063, 3108, 3140, 3144, 3160, 3171, 3192, 3220, 3336, 3363, 3379, 3432, 3435, 3443, 3460, 3480, 3531, 3556, 3588, 3603, 3640, 3732, 3752, 3784, 3795, 3819, 3828, 3832, 3939, 3976, 4008, 4020, 4043, 4171, 4179, 4180, 4216, 4228, 4251, 4260, 4324, 4379, 4420, 4427, 4440, 4452, 4488, 4515, 4516, 4596, 4612, 4683, 4687, 4712, 4740, 4804, 4899, 4939, 4971, 4984, 5115, 5160, 5187, 5195, 5208, 5363, 5380, 5403, 5412, 5428, 5460, 5572, 5668, 5752, 5848, 5860, 5883, 5896, 5907, 5908, 5992, 5995, 6040, 6052, 6099, 6123, 6148, 6195, 6312, 6315, 6328, 6355, 6395, 6420, 6532, 6580, 6595, 6612, 6628, 6708, 6747, 6771, 6792, 6820, 6868, 6923, 6952, 7003, 7035, 7051, 7195, 7288, 7315, 7347, 7368, 7395, 7480, 7491, 7540, 7579, 7588, 7672, 7707, 7747, 7755, 7780, 7795, 7819, 7828, 7843, 7923, 7995, 8008, 8043, 8052, 8083, 8283, 8299, 8308, 8452, 8515, 8547, 8548, 8635, 8643, 8680, 8683, 8715, 8835, 8859, 8932, 8968, 9208, 9219, 9412, 9483, 9507, 9508, 9595, 9640, 9763, 9835, 9867, 9955, 10132, 10168, 10195, 10203, 10227, 10312, 10387, 10420, 10563, 10587, 10635, 10803, 10843, 10948, 10963, 11067, 11092, 11107, 11179, 11203, 11512, 11523, 11563, 11572, 11635, 11715, 11848, 11995, 12027, 12259, 12387, 12523, 12595, 12747, 12772, 12835, 12859, 12868, 13123, 13192, 13195, 13288, 13323, 13363, 13507, 13795, 13819, 13827, 14008, 14155, 14371, 14403, 14547, 14707, 14763, 14995, 15067, 15387, 15403, 15547, 15715, 16027, 16195, 16347, 16531, 16555, 16723, 17227, 17323, 17347, 17427, 17515, 18403, 18715, 18883, 18907, 19147, 19195, 19947, 19987, 20155, 20395, 21403, 21715, 21835, 22243, 22843, 23395, 23587, 24403, 25027, 25267, 27307, 27787, 28963, 31243 17 45 Sloane's A046014 383, 991, 1091, 1571, 1663, 1783, 2531, 3323, 3947, 4339, 4447, 4547, 4651, 5483, 6203, 6379, 6451, 6827, 6907, 7883, 8539, 8731, 9883, 11251, 11443, 12907, 13627, 14083, 14779, 14947, 16699, 17827, 18307, 19963, 21067, 23563, 24907, 25243, 26083, 26107, 27763, 31627, 33427, 36523, 37123 18 150 Sloane's A046015 335, 519, 527, 679, 1135, 1172, 1207, 1383, 1448, 1687, 1691, 1927, 2047, 2051, 2167, 2228, 2291, 2315, 2344, 2644, 2747, 2859, 3035, 3107, 3543, 3544, 3651, 3688, 4072, 4299, 4307, 4568, 4819, 4883, 5224, 5315, 5464, 5492, 5539, 5899, 6196, 6227, 6331, 6387, 6484, 6739, 6835, 7323, 7339, 7528, 7571, 7715, 7732, 7771, 7827, 8152, 8203, 8212, 8331, 8403, 8488, 8507, 8587, 8884, 9123, 9211, 9563, 9627, 9683, 9748, 9832, 10228, 10264, 10347, 10523, 11188, 11419, 11608, 11643, 11683, 11851, 11992, 12067, 12148, 12187, 12235, 12283, 12651, 12723, 12811, 12952, 13227, 13315, 13387, 13747, 13947, 13987, 14163, 14227, 14515, 14667, 14932, 15115, 15243, 16123, 16171, 16387, 16627, 17035, 17131, 17403, 17635, 18283, 18712, 19027, 19123, 19651, 20035, 20827, 21043, 21652, 21667, 21907, 22267, 22443, 22507, 22947, 23347, 23467, 23683, 23923, 24067, 24523, 24667, 24787, 25435, 26587, 26707, 28147, 29467, 32827, 33763, 34027, 34507, 36667, 39307, 40987, 41827, 43387, 48427 19 47 Sloane's A046016 311, 359, 919, 1063, 1543, 1831, 2099, 2339, 2459, 3343, 3463, 3467, 3607, 4019, 4139, 4327, 5059, 5147, 5527, 5659, 6803, 8419, 8923, 8971, 9619, 10891, 11299, 15091, 15331, 16363, 16747, 17011, 17299, 17539, 17683, 19507, 21187, 21211, 21283, 23203, 24763, 26227, 27043, 29803, 31123, 37507, 38707 20 350 Sloane's A046017 455, 615, 776, 824, 836, 920, 1064, 1124, 1160, 1263, 1284, 1460, 1495, 1524, 1544, 1592, 1604, 1652, 1695, 1739, 1748, 1796, 1880, 1887, 1896, 1928, 1940, 1956, 2136, 2247, 2360, 2404, 2407, 2483, 2487, 2532, 2552, 2596, 2603, 2712, 2724, 2743, 2948, 2983, 2987, 3007, 3016, 3076, 3099, 3103, 3124, 3131, 3155, 3219, 3288, 3320, 3367, 3395, 3496, 3512, 3515, 3567, 3655, 3668, 3684, 3748, 3755, 3908, 3979, 4011, 4015, 4024, 4036, 4148, 4264, 4355, 4371, 4395, 4403, 4408, 4539, 4548, 4660, 4728, 4731, 4756, 4763, 4855, 4891, 5019, 5028, 5044, 5080, 5092, 5268, 5331, 5332, 5352, 5368, 5512, 5560, 5592, 5731, 5944, 5955, 5956, 5988, 6051, 6088, 6136, 6139, 6168, 6280, 6339, 6467, 6504, 6648, 6712, 6755, 6808, 6856, 7012, 7032, 7044, 7060, 7096, 7131, 7144, 7163, 7171, 7192, 7240, 7428, 7432, 7467, 7572, 7611, 7624, 7635, 7651, 7667, 7720, 7851, 7876, 7924, 7939, 8067, 8251, 8292, 8296, 8355, 8404, 8472, 8491, 8632, 8692, 8755, 8808, 8920, 8995, 9051, 9124, 9147, 9160, 9195, 9331, 9339, 9363, 9443, 9571, 9592, 9688, 9691, 9732, 9755, 9795, 9892, 9976, 9979, 10027, 10083, 10155, 10171, 10291, 10299, 10308, 10507, 10515, 10552, 10564, 10819, 10888, 11272, 11320, 11355, 11379, 11395, 11427, 11428, 11539, 11659, 11755, 11860, 11883, 11947, 11955, 12019, 12139, 12280, 12315, 12328, 12331, 12355, 12363, 12467, 12468, 12472, 12499, 12532, 12587, 12603, 12712, 12883, 12931, 12955, 12963, 13155, 13243, 13528, 13555, 13588, 13651, 13803, 13960, 14307, 14331, 14467, 14491, 14659, 14755, 14788, 15235, 15268, 15355, 15603, 15688, 15691, 15763, 15883, 15892, 15955, 16147, 16228, 16395, 16408, 16435, 16483, 16507, 16612, 16648, 16683, 16707, 16915, 16923, 17067, 17187, 17368, 17563, 17643, 17763, 17907, 18067, 18163, 18195, 18232, 18355, 18363, 19083, 19443, 19492, 19555, 19923, 20083, 20203, 20587, 20683, 20755, 20883, 21091, 21235, 21268, 21307, 21387, 21508, 21595, 21723, 21763, 21883, 22387, 22467, 22555, 22603, 22723, 23443, 23947, 24283, 24355, 24747, 24963, 25123, 25363, 26635, 26755, 26827, 26923, 27003, 27955, 27987, 28483, 28555, 29107, 29203, 30283, 30787, 31003, 31483, 31747, 31987, 32923, 33163, 34435, 35683, 35995, 36283, 37627, 37843, 37867, 38347, 39187, 39403, 40243, 40363, 40555, 40723, 43747, 47083, 48283, 51643, 54763, 58507 21 85 Sloane's A046018 431, 503, 743, 863, 1931, 2503, 2579, 2767, 2819, 3011, 3371, 4283, 4523, 4691, 5011, 5647, 5851, 5867, 6323, 6691, 7907, 8059, 8123, 8171, 8243, 8387, 8627, 8747, 9091, 9187, 9811, 9859, 10067, 10771, 11731, 12107, 12547, 13171, 13291, 13339, 13723, 14419, 14563, 15427, 16339, 16987, 17107, 17707, 17971, 18427, 18979, 19483, 19531, 19819, 20947, 21379, 22027, 22483, 22963, 23227, 23827, 25603, 26683, 27427, 28387, 28723, 28867, 31963, 32803, 34147, 34963, 35323, 36067, 36187, 39043, 40483, 44683, 46027, 49603, 51283, 52627, 55603, 58963, 59467, 61483 22 139 Sloane's A046019 591, 623, 767, 871, 879, 1076, 1111, 1167, 1304, 1556, 1591, 1639, 1903, 2215, 2216, 2263, 2435, 2623, 2648, 2815, 2863, 2935, 3032, 3151, 3316, 3563, 3587, 3827, 4084, 4115, 4163, 4328, 4456, 4504, 4667, 4811, 5383, 5416, 5603, 5716, 5739, 5972, 6019, 6127, 6243, 6616, 6772, 6819, 7179, 7235, 7403, 7763, 7768, 7899, 8023, 8143, 8371, 8659, 8728, 8851, 8907, 8915, 9267, 9304, 9496, 10435, 10579, 10708, 10851, 11035, 11283, 11363, 11668, 12091, 12115, 12403, 12867, 13672, 14019, 14059, 14179, 14548, 14587, 14635, 15208, 15563, 15832, 16243, 16251, 16283, 16291, 16459, 17147, 17587, 17779, 17947, 18115, 18267, 18835, 18987, 19243, 19315, 19672, 20308, 20392, 22579, 22587, 22987, 24243, 24427, 25387, 25507, 25843, 25963, 26323, 26548, 27619, 28267, 29227, 29635, 29827, 30235, 30867, 31315, 33643, 33667, 34003, 34387, 35347, 41083, 43723, 44923, 46363, 47587, 47923, 49723, 53827, 77683, 85507 23 68 Sloane's A046020 647, 1039, 1103, 1279, 1447, 1471, 1811, 1979, 2411, 2671, 3491, 3539, 3847, 3923, 4211, 4783, 5387, 5507, 5531, 6563, 6659, 6703, 7043, 9587, 9931, 10867, 10883, 12203, 12739, 13099, 13187, 15307, 15451, 16267, 17203, 17851, 18379, 20323, 20443, 20899, 21019, 21163, 22171, 22531, 24043, 25147, 25579, 25939, 26251, 26947, 27283, 28843, 30187, 31147, 31267, 32467, 34843, 35107, 37003, 40627, 40867, 41203, 42667, 43003, 45427, 45523, 47947, 90787 24 510 Sloane's A048925 695, 759, 1191, 1316, 1351, 1407, 1615, 1704, 1736, 1743, 1988, 2168, 2184, 2219, 2372, 2408, 2479, 2660, 2696, 2820, 2824, 2852, 2856, 2915, 2964, 3059, 3064, 3127, 3128, 3444, 3540, 3560, 3604, 3620, 3720, 3864, 3876, 3891, 3899, 3912, 3940, 4063, 4292, 4308, 4503, 4564, 4580, 4595, 4632, 4692, 4715, 4744, 4808, 4872, 4920, 4936, 5016, 5124, 5172, 5219, 5235, 5236, 5252, 5284, 5320, 5348, 5379, 5432, 5448, 5555, 5588, 5620, 5691, 5699, 5747, 5748, 5768, 5828, 5928, 5963, 5979, 6004, 6008, 6024, 6072, 6083, 6132, 6180, 6216, 6251, 6295, 6340, 6411, 6531, 6555, 6699, 6888, 6904, 6916, 7048, 7108, 7188, 7320, 7332, 7348, 7419, 7512, 7531, 7563, 7620, 7764, 7779, 7928, 7960, 7972, 8088, 8115, 8148, 8211, 8260, 8328, 8344, 8392, 8499, 8603, 8628, 8740, 8760, 8763, 8772, 8979, 9028, 9048, 9083, 9112, 9220, 9259, 9268, 9347, 9352, 9379, 9384, 9395, 9451, 9480, 9492, 9652, 9672, 9715, 9723, 9823, 9915, 9928, 9940, 10011, 10059, 10068, 10120, 10180, 10187, 10212, 10248, 10283, 10355, 10360, 10372, 10392, 10452, 10488, 10516, 10612, 10632, 10699, 10740, 10756, 10788, 10792, 10840, 10852, 10923, 11019, 11032, 11139, 11176, 11208, 11211, 11235, 11267, 11307, 11603, 11620, 11627, 11656, 11667, 11748, 11752, 11811, 11812, 11908, 11928, 12072, 12083, 12243, 12292, 12376, 12408, 12435, 12507, 12552, 12628, 12760, 12808, 12820, 12891, 13035, 13060, 13080, 13252, 13348, 13395, 13427, 13444, 13512, 13531, 13539, 13540, 13587, 13611, 13668, 13699, 13732, 13780, 13912, 14035, 14043, 14212, 14235, 14260, 14392, 14523, 14532, 14536, 14539, 14555, 14595, 14611, 14632, 14835, 14907, 14952, 14968, 14980, 15019, 15112, 15267, 15339, 15411, 15460, 15483, 15528, 15555, 15595, 15640, 15652, 15747, 15748, 15828, 15843, 15931, 15940, 15988, 16107, 16132, 16315, 16360, 16468, 16563, 16795, 16827, 16872, 16888, 16907, 16948, 17032, 17043, 17059, 17092, 17283, 17560, 17572, 17620, 17668, 17752, 17812, 17843, 18040, 18052, 18088, 18132, 18148, 18340, 18507, 18568, 18579, 18595, 18627, 18628, 18667, 18763, 18795, 18811, 18867, 18868, 18915, 19203, 19528, 19579, 19587, 19627, 19768, 19803, 19912, 19915, 20260, 20307, 20355, 20427, 20491, 20659, 20692, 20728, 20803, 20932, 20955, 20980, 20995, 21112, 21172, 21352, 21443, 21448, 21603, 21747, 21963, 21988, 22072, 22107, 22180, 22323, 22339, 22803, 22852, 22867, 22939, 23032, 23035, 23107, 23115, 23188, 23235, 23307, 23368, 23752, 23907, 23995, 24115, 24123, 24292, 24315, 24388, 24595, 24627, 24628, 24643, 24915, 24952, 24955, 25048, 25195, 25347, 25467, 25683, 25707, 25732, 25755, 25795, 25915, 25923, 25972, 25987, 26035, 26187, 26395, 26427, 26467, 26643, 26728, 26995, 27115, 27163, 27267, 27435, 27448, 27523, 27643, 27652, 27907, 28243, 28315, 28347, 28372, 28459, 28747, 28891, 29128, 29283, 29323, 29395, 29563, 29659, 29668, 29755, 29923, 30088, 30163, 30363, 30387, 30523, 30667, 30739, 30907, 30955, 30979, 31252, 31348, 31579, 31683, 31795, 31915, 32008, 32043, 32155, 32547, 32635, 32883, 33067, 33187, 33883, 34203, 34363, 34827, 34923, 36003, 36043, 36547, 36723, 36763, 36883, 37227, 37555, 37563, 38227, 38443, 38467, 39603, 39643, 39787, 40147, 40195, 40747, 41035, 41563, 42067, 42163, 42267, 42387, 42427, 42835, 43483, 44947, 45115, 45787, 46195, 46243, 46267, 47203, 47443, 47707, 48547, 49107, 49267, 49387, 49987, 50395, 52123, 52915, 54307, 55867, 56947, 57523, 60523, 60883, 61147, 62155, 62203, 63043, 64800, 79363, 84043, 84547

The table below gives lists of Positive fundamental discriminants having small class numbers , corresponding to Real quadratic fields. All Positive Squarefree values of (for which the Kronecker Symbol is defined) are included.

 1 5, 13, 17, 21, 29, 37, 41, 53, 57, 61, 69, 73, 77 2 65

The Positive for which is given by Sloane's A014539.

See also Class Number Formula, Dirichlet L-Series, Discriminant (Binary Quadratic Form), Gauss's Class Number Conjecture, Gauss's Class Number Problem, Heegner Number, Ideal, j-Function

References

Arno, S. The Imaginary Quadratic Fields of Class Number 4.'' Acta Arith. 40, 321-334, 1992.

Arno, S.; Robinson, M. L.; and Wheeler, F. S. Imaginary Quadratic Fields with Small Odd Class Number.'' http://www.math.uiuc.edu/Algebraic-Number-Theory/0009/.

Buell, D. A. Small Class Numbers and Extreme Values of -Functions of Quadratic Fields.'' Math. Comput. 139, 786-796, 1977.

Cohen, H. A Course in Computational Algebraic Number Theory. New York: Springer-Verlag, 1993.

Cohn, H. Advanced Number Theory. New York: Dover, pp. 163 and 234, 1980.

Cox, D. A. Primes of the Form : Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.

Davenport, H. Dirichlet's Class Number Formula.'' Ch. 6 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43-53, 1980.

Iyanaga, S. and Kawada, Y. (Eds.). Class Numbers of Algebraic Number Fields.'' Appendix B, Table 4 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1494-1496, 1980.

Montgomery, H. and Weinberger, P. Notes on Small Class Numbers.'' Acta. Arith. 24, 529-542, 1974.

Sloane, N. J. A. Sequences A014539, A038552, A046125, and A003657/M2332 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Stark, H. M. A Complete Determination of the Complex Quadratic Fields of Class Number One.'' Michigan Math. J. 14, 1-27, 1967.

Stark, H. M. On Complex Quadratic Fields with Class Number Two.'' Math. Comput. 29, 289-302, 1975.

Wagner, C. Class Number 5, 6, and 7.'' Math. Comput. 65, 785-800, 1996.

Weisstein, E. W. Class Numbers.'' Mathematica notebook ClassNumbers.m.