Class Number

For any Ideal , there is an Ideal such that

(1)

A finite series giving exactly the class number of a Ring is known as a Class Number Formula. A ClassNumber 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)

(3)

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

(4)

(5)

(6)

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

(7)

Wagner (1996) shows that class number satisfies the Inequality

(8)

The Mathematica (Wolfram Research, Champaign, IL) function NumberTheory`NumberTheoryFunctions`ClassNumber[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 beensolved for and Odd . Gauß conjectured that the class number of anImaginary quadratic field with Discriminant tendsto 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 numberof ``fundamental'' values of with a given class number , where ``fundamental'' means that is not divisible byany Square Number such that . For example, although , is not a fundamentaldiscriminant since and . Even values have been computed byWeisstein. 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 at50000, 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 toReal quadratic fields. All Positive Squarefree values of (for which theKronecker 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.

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, andA003657/M2332in ``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.

• Disc
• Disconnected Form
• Disconnectivity
• Discontinuity
• Discontinuous
• Discordant Permutation
• Discrepancy Theorem
• Discrete Distribution
• Discrete Fourier Transform
• Discrete Mathematics
• Discrete Set
• Discrete Uniform Distribution
• Discriminant
• Discriminant (Binary Quadratic Form)
• Discriminant (Elliptic Curve)
• Discriminant (Metric)
• Discriminant (Module)
• Discriminant (Polynomial)
• Discriminant (Quadratic Curve)
• Discriminant (Quadratic Form)
• Discriminant (Second Derivative Test)
• Disdyakis Dodecahedron
• Disdyakis Triacontahedron
• Disjoint
• Disjunction