Cyclic Number - Other Numeric Bases

Other Numeric Bases

Using the above technique, cyclic numbers can be found in other numeric bases. (Note that not all of these follow the second rule (all successive multiples being cyclic permutations) listed in the Special Cases section above)

In binary, the sequence of cyclic numbers begins:

01
0011
0001011101
000100111011
000011010111100101

In ternary:

0121
010212
0011202122110201
001102100221120122
0002210102011122200121202111

In octal:

25
1463
0564272135
0215173454106475626043236713
0115220717545336140465103476625570602324416373126743

In duodecimal:

2497
186A35
08579214B36429A7

In Base 24:

3A6LDH
248HAMKF6D
1L795CN3GEJB
19M45FCGNE2KJ8B7

Note that in ternary (b = 3), the case p = 2 yields 1 as a cyclic number. While single digits may be considered trivial cases, it may be useful for completeness of the theory to consider them only when they are generated in this way.

It can be shown that no cyclic numbers (other than trivial single digits) exist in any numeric base which is a perfect square; thus there are no cyclic numbers in hexadecimal.

Read more about this topic:  Cyclic Number

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