This site is supported by donations to The OEIS Foundation.

Class numbers

From OeisWiki
Jump to: navigation, search

This article is under construction.

Please do not rely on any information it contains.



This article page is a stub, please help by expanding it.

The class number of the ring of algebraic integers of tells how far from or how close that ring is to being a unique factorization domain ( is some algebraic integer).

A class number of 1 means the domain is a unique factorization domain (UFD, not necessarily Euclidean, however). A class number of 2 means the domain is not a UFD, but does guarantee that whenever distinct factorizations a particular number in that domain may have, they each involve as many non-unit irreducible factors. A class number of 3 or higher means the domain is not a UFD and there may be no guarantees of similarities between distinct factorizations of a particular number in that domain.

Class numbers have been studied most thoroughly in the case of , where is a squarefree integer, positive or negative.

A-number
1 −163, −67, −43, −19, −11, −7, −3, −2, −1, 1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 29, 31, ... A061574
2 −5, −6, −10, −13, −15, −22, −35, −37, −51, −58, −91, −115, −123, −187, −235, −267, −403, −427 A005847
10, 15, 26, 30, 34, 35, 39, 42, 51, 55, 58, 65, 66, 70, 74, 78, 85, 87, 91, 95, 102, 105, 106, 110, ... A029702
3 −23, −31, −59, −83, −107, −139, −211, −283, −307, −331, −379, −499, −547, −643, −883, −907 A006203
79, 142, 223, 229, 254, 257, 321, 326, 359, 443, 469, 473, 659, 733, 761, 839, 934, 993, 1091, ... A029703
4 −14, −17, −21, −30, −33, −34, −39, −42, −46, −55, −57, −70, −73, −78, −82, −85, −93, −97, −102, ...
(contains only thirty-five more numbers)
A046085
82, 130, 145, 170, 195, 210, 219, 231, 255, 274, 290, 291, 322, 323, 330, 370, 390, 410, 434, 435, ... A029704

Notes: Sequences consisting only of negative numbers are shown multiplied by −1 in their respective entries. Ellipses indicate sequences not known to be finite unless otherwise noted.