1760:
255:) have class number one. In other words, all cubic fields of discriminant between 0 and 1944 (inclusively) have class number one. The next totally real cubic field (of discriminant 1957) has class number two. The polynomials defining the totally real cubic fields that have discriminants less than 500 with class number one are:
1263:
A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken
Yamamura and is available on pages 915–919 of his article on the subject. Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of
423:
All complex cubic fields with discriminant greater than −500 have class number one, except the fields with discriminants −283, −331 and −491 which have class number 2. The real root of the polynomial for −23 is the reciprocal of the
1175:. Weber showed that these fields have odd class number. In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 10, and later improved this bound to 10. These fields are the
201:) does have class number 1 is conjectured to be nonzero, and in fact close to 76%, however it is not even known whether there are infinitely many real quadratic fields with class number 1.
1357:
1788:
1676:
1566:
1810:
1698:
1588:
2048:
Louboutin, Stéphane; Okazaki, Ryotaro (1994), "Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one",
143:
2*, 3, 5*, 6, 7, 11, 13*, 14, 17*, 19, 21, 22, 23, 29*, 31, 33, 37*, 38, 41*, 43, 46, 47, 53*, 57, 59, 61*, 62, 67, 69, 71, 73*, 77, 83, 86, 89*, 93, 94, 97*, ...
167:
Despite what would appear to be the case for these small values, not all prime numbers that are congruent to 1 modulo 4 appear on this list, notably the fields
1505:
252:
136:
1248:
conjectured that there are finitely many CM fields of class number 1. He showed that there are finitely many of a fixed degree. Shortly thereafter,
2102:
1471:
1431:, Noordwijkerhout 1983, Proc. 13th Journées Arithmétiques, ed. H. Jager, Lect. Notes in Math. 1068, Springer-Verlag, 1984, pp. 33—62
55:
2094:
2137:
188:= 257 both have class number greater than 1 (in fact equal to 3 in both cases). The density of such primes for which
1237:
1171:
is a positive integer) are known to have class number 1 for n≤8, and it is conjectured that they have class number 1 for all
432:. The polynomials defining the complex cubic fields that have class number one and discriminant greater than −500 are:
2142:
51:
1260:
showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1.
1232:
Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field
1283:
1118:
1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84.
1929:
1872:
1327:
1976:
1455:
71:
47:
2004:
Yamamura, Ken (1994), "The determination of the imaginary abelian number fields with class number one",
1526:
J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem,
1764:
1652:
1542:
2013:
1938:
1881:
1323:
1278:
1273:
109:
1500:"Sequence A005848 (Cyclotomic fields with class number 1 (or with unique factorization).)"
94:
1793:
1681:
1571:
2031:
1954:
1907:
1733:
1623:
1389:
1241:
156:
2080:
23:
It is believed that there are infinitely many such number fields, but this has not been proven.
226:−1, −2, −3, −7, −11, −19, −43, −67, −163.
2098:
1833:
1717:
1607:
1467:
1443:
1381:
1201:
have no prime factor less than 10. Coates has raised the question of whether, for all primes
429:
36:
2116:
2057:
2021:
1985:
1946:
1897:
1889:
1849:
1823:
1741:
1709:
1631:
1599:
1477:
1397:
1373:
1095:
40:
2112:
1845:
1729:
1619:
2120:
2108:
1853:
1841:
1745:
1725:
1649:
Fukuda, Takashi; Komatsu, Keiichi (2011). "Weber's class number problem in the cyclotomic
1635:
1615:
1539:
Fukuda, Takashi; Komatsu, Keiichi (2009). "Weber's class number problem in the cyclotomic
1481:
1463:
1401:
2017:
1942:
1885:
1190:. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic
1971:
1924:
1257:
1249:
210:
2131:
1958:
1911:
1737:
425:
17:
1627:
1867:
1603:
1393:
1377:
1245:
46:
Thus, a number field has class number 1 if and only if its ring of integers is a
1495:
241:
1990:
1902:
1713:
1837:
1828:
1721:
1611:
1385:
2062:
1227:
2035:
1950:
1893:
1927:(1975), "Some analytic estimates of class numbers and discriminants",
2026:
1974:(2001), "Class numbers of CM-fields with solvable normal closure",
1527:
218:
has class number 1 exactly for the 9 following negative values of
1462:. Graduate Texts in Mathematics. Vol. 83 (2nd ed.).
1870:(1974), "Some effective cases of the Brauer–Siegel theorem",
428:(negated), while that for −31 is the reciprocal of the
1499:
231:(By definition, these also all have narrow class number 1.)
160:
131:
1429:
Heuristics on class groups of number fields, Number Theory
1796:
1767:
1684:
1655:
1574:
1545:
1418:, GTM 138, Springer Verlag (1993), Appendix B2, p.507
1330:
125:
has class number 1 for the following values of
35:
of a number field is by definition the order of the
1324:"Explicit computations of Hilbert modular forms on
251:The first 60 totally real cubic fields (ordered by
1804:
1782:
1692:
1670:
1582:
1560:
1351:
1416:A Course in Computational Algebraic Number Theory
1155:On the other hand, the maximal real subfields
2090:Grundlehren der mathematischen Wissenschaften
1159:(cos(2Ď€/2)) of the 2-power cyclotomic fields
8:
2088:
1127:congruent to 2 modulo 4 are redundant since
1100:The following is a complete list of thirty
1761:"A class number problem in the cyclotomic
2061:
2025:
1989:
1901:
1827:
1798:
1797:
1795:
1774:
1770:
1769:
1766:
1686:
1685:
1683:
1662:
1658:
1657:
1654:
1576:
1575:
1573:
1552:
1548:
1547:
1544:
1506:On-Line Encyclopedia of Integer Sequences
1352:{\displaystyle \mathbb {Q} ({\sqrt {5}})}
1339:
1332:
1331:
1329:
1252:showed that there are only finitely many
1310:
1295:
1439:
1437:
1256:CM fields of class number 1. In 2001,
1305:
1303:
1301:
1299:
7:
1444:Tables available at Pari source code
1084:− 3 (discriminant −499)
1046:− 6 (discriminant −472)
1036:− 3 (discriminant −460)
1022:− 8 (discriminant −459)
998:− 8 (discriminant −440)
974:− 4 (discriminant −436)
964:− 8 (discriminant −431)
944:− 5 (discriminant −419)
934:− 2 (discriminant −411)
920:− 4 (discriminant −379)
892:− 2 (discriminant −364)
868:− 3 (discriminant −351)
830:− 3 (discriminant −327)
816:− 4 (discriminant −324)
792:− 3 (discriminant −300)
764:− 3 (discriminant −255)
754:− 3 (discriminant −247)
744:− 6 (discriminant −244)
734:− 3 (discriminant −243)
728:− 3 (discriminant −239)
708:− 2 (discriminant −216)
698:− 2 (discriminant −212)
684:− 3 (discriminant −211)
674:− 3 (discriminant −204)
646:− 1 (discriminant −199)
632:− 3 (discriminant −175)
604:− 2 (discriminant −152)
590:− 2 (discriminant −140)
566:− 1 (discriminant −135)
556:− 2 (discriminant −116)
546:− 2 (discriminant −108)
540:− 2 (discriminant −107)
526:− 2 (discriminant −104)
502:− 2 (discriminant −83)
488:− 2 (discriminant −76)
478:− 1 (discriminant −59)
454:− 1 (discriminant −31)
14:
1460:Introduction to Cyclotomic Fields
56:fundamental theorem of arithmetic
1783:{\displaystyle \mathbb {Z} _{3}}
1671:{\displaystyle \mathbb {Z} _{2}}
1561:{\displaystyle \mathbb {Z} _{2}}
1205:, every layer of the cyclotomic
159:is also 1 (see related sequence
1528:https://arxiv.org/abs/1405.1094
1309:Chapter I, section 6, p. 37 of
1604:10.1080/10586458.2009.10128896
1378:10.1080/10586458.2005.10128939
1346:
1336:
16:This is an incomplete list of
1:
1179:-th layers of the cyclotomic
1074:+ 3 (discriminant −492)
1060:+ 2 (discriminant −484)
1012:+ 8 (discriminant −451)
988:+ 5 (discriminant −439)
954:+ 8 (discriminant −424)
906:+ 3 (discriminant −367)
882:+ 7 (discriminant −356)
858:+ 4 (discriminant −339)
844:+ 1 (discriminant −335)
806:+ 2 (discriminant −307)
778:+ 5 (discriminant −268)
718:+ 3 (discriminant −231)
660:+ 2 (discriminant −200)
618:+ 3 (discriminant −172)
580:+ 2 (discriminant −139)
1805:{\displaystyle \mathbb {Q} }
1693:{\displaystyle \mathbb {Q} }
1583:{\displaystyle \mathbb {Q} }
1427:H. Cohen and H. W. Lenstra,
516:+ 1 (discriminant −87)
468:+ 1 (discriminant −44)
444:+ 1 (discriminant −23)
413:− 1 (discriminant 473)
389:− 1 (discriminant 404)
319:− 1 (discriminant 229)
309:− 1 (discriminant 169)
117:is called real quadratic if
1759:Morisawa, Takayuki (2009).
281:− 1 (discriminant 81)
52:unique factorization domain
2159:
2085:Algebraische Zahlentheorie
2006:Mathematics of Computation
1496:Sloane, N. J. A.
1225:
1093:
239:
208:
205:Imaginary quadratic fields
107:
69:
2093:. Vol. 322. Berlin:
1714:10.1142/S1793042111004782
1322:Dembélé, Lassina (2005).
1240:quadratic extension of a
1930:Inventiones Mathematicae
1873:Inventiones Mathematicae
247:Totally real cubic field
2138:Algebraic number theory
1991:10.1023/A:1017589432526
1456:Washington, Lawrence C.
66:Quadratic number fields
2089:
1977:Compositio Mathematica
1829:10.3836/tjm/1264170249
1806:
1784:
1694:
1672:
1584:
1562:
1353:
1123:(Note that values of
1114:) has class number 1:
403:+ 4 (discriminant 469)
375:+ 7 (discriminant 361)
361:+ 1 (discriminant 321)
347:+ 2 (discriminant 316)
333:+ 3 (discriminant 257)
295:+ 1 (discriminant 148)
76:These are of the form
72:Quadratic number field
48:principal ideal domain
2063:10.4064/aa-67-1-47-62
1807:
1785:
1702:Int. J. Number Theory
1695:
1673:
1585:
1563:
1354:
1284:Brauer–Siegel theorem
271:+ 1 (discriminant 49)
104:Real quadratic fields
20:with class number 1.
1794:
1765:
1682:
1653:
1572:
1543:
1328:
1279:Class number formula
1274:Class number problem
1218:has class number 1.
1104:for which the field
110:Real quadratic field
62:has class number 1.
2143:Field (mathematics)
2018:1994MaCom..62..899Y
1943:1975InMat..29..275O
1886:1974InMat..23..135S
419:Complex cubic field
157:narrow class number
95:square-free integer
1951:10.1007/bf01389854
1903:10338.dmlcz/120573
1894:10.1007/bf01405166
1802:
1780:
1690:
1668:
1580:
1558:
1349:
1242:totally real field
151: = 100)
2104:978-3-540-65399-8
1509:. OEIS Foundation
1344:
1238:totally imaginary
1152:
1090:Cyclotomic fields
430:supergolden ratio
232:
164:
37:ideal class group
2150:
2124:
2092:
2081:Neukirch, JĂĽrgen
2067:
2066:
2065:
2050:Acta Arithmetica
2045:
2039:
2038:
2029:
2012:(206): 899–921,
2001:
1995:
1994:
1993:
1968:
1962:
1961:
1921:
1915:
1914:
1905:
1864:
1858:
1857:
1831:
1811:
1809:
1808:
1803:
1801:
1789:
1787:
1786:
1781:
1779:
1778:
1773:
1756:
1750:
1749:
1708:(6): 1627–1635.
1699:
1697:
1696:
1691:
1689:
1677:
1675:
1674:
1669:
1667:
1666:
1661:
1646:
1640:
1639:
1589:
1587:
1586:
1581:
1579:
1567:
1565:
1564:
1559:
1557:
1556:
1551:
1536:
1530:
1524:
1518:
1517:
1515:
1514:
1492:
1486:
1485:
1466:. Theorem 11.1.
1452:
1446:
1441:
1432:
1425:
1419:
1412:
1406:
1405:
1363:
1358:
1356:
1355:
1350:
1345:
1340:
1335:
1319:
1313:
1307:
1264:class number 1.
1122:
1096:Cyclotomic field
230:
200:
199:
179:
178:
154:
147:(complete until
134:
92:
91:
41:ring of integers
2158:
2157:
2153:
2152:
2151:
2149:
2148:
2147:
2128:
2127:
2105:
2095:Springer-Verlag
2079:
2076:
2071:
2070:
2047:
2046:
2042:
2027:10.2307/2153549
2003:
2002:
1998:
1972:Murty, V. Kumar
1970:
1969:
1965:
1925:Odlyzko, Andrew
1923:
1922:
1918:
1866:
1865:
1861:
1792:
1791:
1768:
1763:
1762:
1758:
1757:
1753:
1680:
1679:
1656:
1651:
1650:
1648:
1647:
1643:
1570:
1569:
1546:
1541:
1540:
1538:
1537:
1533:
1525:
1521:
1512:
1510:
1494:
1493:
1489:
1474:
1464:Springer-Verlag
1454:
1453:
1449:
1442:
1435:
1426:
1422:
1413:
1409:
1361:
1326:
1325:
1321:
1320:
1316:
1308:
1297:
1292:
1270:
1230:
1224:
1213:
1196:
1185:
1166:
1146:
1136:
1113:
1098:
1092:
1087:
421:
416:
249:
244:
238:
213:
207:
195:
193:
174:
172:
130:
112:
106:
87:
85:
74:
68:
29:
12:
11:
5:
2156:
2154:
2146:
2145:
2140:
2130:
2129:
2126:
2125:
2103:
2075:
2072:
2069:
2068:
2040:
1996:
1984:(3): 273–287,
1963:
1937:(3): 275–286,
1916:
1880:(2): 135–152,
1859:
1822:(2): 549–558.
1800:
1790:-extension of
1777:
1772:
1751:
1688:
1678:-extension of
1665:
1660:
1641:
1598:(2): 213–222.
1578:
1568:-extension of
1555:
1550:
1531:
1519:
1487:
1472:
1447:
1433:
1420:
1407:
1372:(4): 457–466.
1348:
1343:
1338:
1334:
1314:
1294:
1293:
1291:
1288:
1287:
1286:
1281:
1276:
1269:
1266:
1258:V. Kumar Murty
1250:Andrew Odlyzko
1226:Main article:
1223:
1220:
1214:-extension of
1209:
1197:-extension of
1194:
1186:-extension of
1183:
1164:
1142:
1137:) =
1132:
1120:
1119:
1109:
1094:Main article:
1091:
1088:
1086:
1085:
1075:
1061:
1047:
1037:
1023:
1013:
999:
989:
975:
965:
955:
945:
935:
921:
907:
893:
883:
869:
859:
845:
831:
817:
807:
793:
779:
765:
755:
745:
735:
729:
719:
709:
699:
685:
675:
661:
647:
633:
619:
605:
591:
581:
567:
557:
547:
541:
527:
517:
503:
489:
479:
469:
455:
445:
434:
420:
417:
415:
414:
404:
390:
376:
362:
348:
334:
320:
310:
296:
282:
272:
257:
248:
245:
240:Main article:
237:
234:
228:
227:
211:Heegner number
209:Main article:
206:
203:
145:
144:
108:Main article:
105:
102:
70:Main article:
67:
64:
28:
25:
13:
10:
9:
6:
4:
3:
2:
2155:
2144:
2141:
2139:
2136:
2135:
2133:
2122:
2118:
2114:
2110:
2106:
2100:
2096:
2091:
2086:
2082:
2078:
2077:
2073:
2064:
2059:
2055:
2051:
2044:
2041:
2037:
2033:
2028:
2023:
2019:
2015:
2011:
2007:
2000:
1997:
1992:
1987:
1983:
1979:
1978:
1973:
1967:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1931:
1926:
1920:
1917:
1913:
1909:
1904:
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1874:
1869:
1868:Stark, Harold
1863:
1860:
1855:
1851:
1847:
1843:
1839:
1835:
1830:
1825:
1821:
1817:
1816:Tokyo J. Math
1813:
1775:
1755:
1752:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1663:
1645:
1642:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1553:
1535:
1532:
1529:
1523:
1520:
1508:
1507:
1501:
1497:
1491:
1488:
1483:
1479:
1475:
1473:0-387-94762-0
1469:
1465:
1461:
1457:
1451:
1448:
1445:
1440:
1438:
1434:
1430:
1424:
1421:
1417:
1411:
1408:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1360:
1341:
1318:
1315:
1312:
1311:Neukirch 1999
1306:
1304:
1302:
1300:
1296:
1289:
1285:
1282:
1280:
1277:
1275:
1272:
1271:
1267:
1265:
1261:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1229:
1221:
1219:
1217:
1212:
1208:
1204:
1200:
1193:
1189:
1182:
1178:
1174:
1170:
1162:
1158:
1153:
1150:
1145:
1140:
1135:
1130:
1126:
1117:
1116:
1115:
1112:
1107:
1103:
1097:
1089:
1083:
1079:
1076:
1073:
1069:
1065:
1062:
1059:
1055:
1051:
1048:
1045:
1041:
1038:
1035:
1031:
1027:
1024:
1021:
1017:
1014:
1011:
1007:
1003:
1000:
997:
993:
990:
987:
983:
979:
976:
973:
969:
966:
963:
959:
956:
953:
949:
946:
943:
939:
936:
933:
929:
925:
922:
919:
915:
911:
908:
905:
901:
897:
894:
891:
887:
884:
881:
877:
873:
870:
867:
863:
860:
857:
853:
849:
846:
843:
839:
835:
832:
829:
825:
821:
818:
815:
811:
808:
805:
801:
797:
794:
791:
787:
783:
780:
777:
773:
769:
766:
763:
759:
756:
753:
749:
746:
743:
739:
736:
733:
730:
727:
723:
720:
717:
713:
710:
707:
703:
700:
697:
693:
689:
686:
683:
679:
676:
673:
669:
665:
662:
659:
655:
651:
648:
645:
641:
637:
634:
631:
627:
623:
620:
617:
613:
609:
606:
603:
599:
595:
592:
589:
585:
582:
579:
575:
571:
568:
565:
561:
558:
555:
551:
548:
545:
542:
539:
535:
531:
528:
525:
521:
518:
515:
511:
507:
504:
501:
497:
493:
490:
487:
483:
480:
477:
473:
470:
467:
463:
459:
456:
453:
449:
446:
443:
439:
436:
435:
433:
431:
427:
426:plastic ratio
418:
412:
408:
405:
402:
398:
394:
391:
388:
384:
380:
377:
374:
370:
366:
363:
360:
356:
352:
349:
346:
342:
338:
335:
332:
328:
324:
321:
318:
314:
311:
308:
304:
300:
297:
294:
290:
286:
283:
280:
276:
273:
270:
266:
262:
259:
258:
256:
254:
246:
243:
235:
233:
225:
224:
223:
221:
217:
212:
204:
202:
198:
191:
187:
183:
177:
170:
165:
162:
158:
152:
150:
142:
141:
140:
138:
133:
128:
124:
120:
116:
111:
103:
101:
99:
96:
90:
83:
80: =
79:
73:
65:
63:
61:
57:
53:
49:
44:
42:
38:
34:
26:
24:
21:
19:
18:number fields
2084:
2056:(1): 47–62,
2053:
2049:
2043:
2009:
2005:
1999:
1981:
1975:
1966:
1934:
1928:
1919:
1877:
1871:
1862:
1819:
1815:
1754:
1705:
1701:
1644:
1595:
1591:
1534:
1522:
1511:. Retrieved
1503:
1490:
1459:
1450:
1428:
1423:
1415:
1410:
1369:
1365:
1317:
1262:
1253:
1246:Harold Stark
1233:
1231:
1215:
1210:
1206:
1202:
1198:
1191:
1187:
1180:
1176:
1172:
1168:
1160:
1156:
1154:
1148:
1143:
1138:
1133:
1128:
1124:
1121:
1110:
1105:
1101:
1099:
1081:
1077:
1071:
1067:
1063:
1057:
1053:
1049:
1043:
1039:
1033:
1029:
1025:
1019:
1015:
1009:
1005:
1001:
995:
991:
985:
981:
977:
971:
967:
961:
957:
951:
947:
941:
937:
931:
927:
923:
917:
913:
909:
903:
899:
895:
889:
885:
879:
875:
871:
865:
861:
855:
851:
847:
841:
837:
833:
827:
823:
819:
813:
809:
803:
799:
795:
789:
785:
781:
775:
771:
767:
761:
757:
751:
747:
741:
737:
731:
725:
721:
715:
711:
705:
701:
695:
691:
687:
681:
677:
671:
667:
663:
657:
653:
649:
643:
639:
635:
629:
625:
621:
615:
611:
607:
601:
597:
593:
587:
583:
577:
573:
569:
563:
559:
553:
549:
543:
537:
533:
529:
523:
519:
513:
509:
505:
499:
495:
491:
485:
481:
475:
471:
465:
461:
457:
451:
447:
441:
437:
422:
410:
406:
400:
396:
392:
386:
382:
378:
372:
368:
364:
358:
354:
350:
344:
340:
336:
330:
326:
322:
316:
312:
306:
302:
298:
292:
288:
284:
278:
274:
268:
264:
260:
253:discriminant
250:
236:Cubic fields
229:
219:
215:
214:
196:
189:
185:
181:
175:
168:
166:
153:
148:
146:
126:
122:
118:
114:
113:
97:
88:
81:
77:
75:
59:
50:(and thus a
45:
33:class number
32:
30:
22:
15:
1244:. In 1974,
242:Cubic field
2132:Categories
2121:0956.11021
2074:References
1854:1205.11116
1746:1226.11119
1636:1189.11033
1513:2024-03-20
1482:0966.11047
1414:H. Cohen,
1402:1152.11328
184:= 229 and
129:(sequence
58:says that
27:Definition
1959:119348804
1912:119482000
1838:0387-3870
1738:121397082
1722:1793-7310
1612:1058-6458
1592:Exp. Math
1386:1058-6458
1366:Exp. Math
1236:, i.e. a
1222:CM fields
1167:) (where
1042:− 5
1008:− 5
984:− 2
940:− 4
826:− 2
812:− 3
788:− 3
774:− 3
680:− 2
600:− 2
484:− 2
409:− 5
399:− 5
385:− 5
371:− 6
357:− 4
343:− 4
329:− 4
315:− 4
305:− 4
291:− 3
277:− 3
267:− 2
163:in OEIS).
93:), for a
2083:(1999).
1628:31421633
1458:(1997).
1268:See also
1228:CM field
1151:is odd.)
1066:−
1052:−
1028:−
1004:−
980:−
960:−
950:−
926:−
912:−
898:−
874:−
854:−
850:−
836:−
822:−
798:−
784:−
770:−
760:−
724:−
714:−
690:−
666:−
652:−
638:−
624:−
614:−
610:−
596:−
572:−
552:−
532:−
522:−
508:−
494:−
460:−
440:−
395:−
381:−
367:−
353:−
339:−
325:−
301:−
287:−
263:−
121:> 0.
2113:1697859
2036:2153549
2014:Bibcode
1939:Bibcode
1882:Bibcode
1846:2589962
1730:2835816
1620:2549691
1498:(ed.).
1394:9088028
1147:) when
194:√
173:√
161:A003655
155:*: The
135:in the
132:A003172
86:√
54:). The
39:of its
2119:
2111:
2101:
2034:
1957:
1910:
1852:
1844:
1836:
1744:
1736:
1728:
1720:
1700:III".
1634:
1626:
1618:
1610:
1480:
1470:
1400:
1392:
1384:
1254:Galois
180:) for
2032:JSTOR
1955:S2CID
1908:S2CID
1734:S2CID
1624:S2CID
1390:S2CID
1362:(PDF)
1290:Notes
2099:ISBN
1834:ISSN
1718:ISSN
1608:ISSN
1504:The
1468:ISBN
1382:ISSN
137:OEIS
31:The
2117:Zbl
2058:doi
2022:doi
1986:doi
1982:127
1947:doi
1898:hdl
1890:doi
1850:Zbl
1824:doi
1742:Zbl
1710:doi
1632:Zbl
1600:doi
1590:".
1478:Zbl
1398:Zbl
1374:doi
1080:+ 4
1070:+ 3
1056:+ 4
1032:+ 5
1018:+ 3
994:+ 2
930:+ 5
902:+ 2
888:+ 4
864:+ 3
840:+ 4
802:+ 3
704:+ 3
694:+ 4
656:+ 2
642:+ 4
628:+ 2
586:+ 2
562:+ 3
536:+ 3
512:+ 2
474:+ 2
139:):
2134::
2115:.
2109:MR
2107:.
2097:.
2087:.
2054:67
2052:,
2030:,
2020:,
2010:62
2008:,
1980:,
1953:,
1945:,
1935:29
1933:,
1906:,
1896:,
1888:,
1878:23
1876:,
1848:.
1842:MR
1840:.
1832:.
1820:32
1818:.
1814:.
1740:.
1732:.
1726:MR
1724:.
1716:.
1704:.
1630:.
1622:.
1616:MR
1614:.
1606:.
1596:18
1594:.
1502:.
1476:.
1436:^
1396:.
1388:.
1380:.
1370:14
1368:.
1364:.
1298:^
1163:(ζ
1141:(ζ
1134:2n
1131:(ζ
1108:(ζ
970:+
916:+
878:+
750:+
740:+
670:+
576:+
498:+
464:+
450:+
222::
100:.
43:.
2123:.
2060::
2024::
2016::
1988::
1949::
1941::
1900::
1892::
1884::
1856:.
1826::
1812:"
1799:Q
1776:3
1771:Z
1748:.
1712::
1706:7
1687:Q
1664:2
1659:Z
1638:.
1602::
1577:Q
1554:2
1549:Z
1516:.
1484:.
1404:.
1376::
1359:"
1347:)
1342:5
1337:(
1333:Q
1234:K
1216:Q
1211:p
1207:Z
1203:p
1199:Q
1195:3
1192:Z
1188:Q
1184:2
1181:Z
1177:n
1173:n
1169:n
1165:2
1161:Q
1157:Q
1149:n
1144:n
1139:Q
1129:Q
1125:n
1111:n
1106:Q
1102:n
1082:x
1078:x
1072:x
1068:x
1064:x
1058:x
1054:x
1050:x
1044:x
1040:x
1034:x
1030:x
1026:x
1020:x
1016:x
1010:x
1006:x
1002:x
996:x
992:x
986:x
982:x
978:x
972:x
968:x
962:x
958:x
952:x
948:x
942:x
938:x
932:x
928:x
924:x
918:x
914:x
910:x
904:x
900:x
896:x
890:x
886:x
880:x
876:x
872:x
866:x
862:x
856:x
852:x
848:x
842:x
838:x
834:x
828:x
824:x
820:x
814:x
810:x
804:x
800:x
796:x
790:x
786:x
782:x
776:x
772:x
768:x
762:x
758:x
752:x
748:x
742:x
738:x
732:x
726:x
722:x
716:x
712:x
706:x
702:x
696:x
692:x
688:x
682:x
678:x
672:x
668:x
664:x
658:x
654:x
650:x
644:x
640:x
636:x
630:x
626:x
622:x
616:x
612:x
608:x
602:x
598:x
594:x
588:x
584:x
578:x
574:x
570:x
564:x
560:x
554:x
550:x
544:x
538:x
534:x
530:x
524:x
520:x
514:x
510:x
506:x
500:x
496:x
492:x
486:x
482:x
476:x
472:x
466:x
462:x
458:x
452:x
448:x
442:x
438:x
411:x
407:x
401:x
397:x
393:x
387:x
383:x
379:x
373:x
369:x
365:x
359:x
355:x
351:x
345:x
341:x
337:x
331:x
327:x
323:x
317:x
313:x
307:x
303:x
299:x
293:x
289:x
285:x
279:x
275:x
269:x
265:x
261:x
220:d
216:K
197:d
192:(
190:Q
186:d
182:d
176:d
171:(
169:Q
149:d
127:d
123:K
119:d
115:K
98:d
89:d
84:(
82:Q
78:K
60:Q
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.