查看“︁循环数”︁的源代码

'''循环数'''（{{lang-en|cyclic number}}），是一类特殊的[[整数]]，其包含的各个数字的循环排列恰为该数的连续倍数 ; 一個n位的循环数的性質是它乘以1至n都是各个数字的循环排列 , 乘以(n+1)會出現[[純位數]] , 純位數每個位都是9。例如，最知名的循环数是[[142857]]：
:142857 &times; 1 = 142857
:142857 &times; 2 = 285714
:142857 &times; 3 = 428571
:142857 &times; 4 = 571428
:142857 &times; 5 = 714285
:142857 &times; 6 = 857142

'''乘以7出現純位數'''
:142857 &times; 7 = 999999

另一例子為(0)588235294117647
:588235294117647 &times; 1 = 588235294117647
:588235294117647 &times; 2 = 1176470588235294
:588235294117647 &times; 3 = 1764705882352941
:588235294117647 &times; 4 = 2352941176470588
:588235294117647 &times; 5 = 2941176470588235
:588235294117647 &times; 6 = 3529411764705882
:588235294117647 &times; 7 = 4117647058823529
:588235294117647 &times; 8 = 4705882352941176
:588235294117647 &times; 9 = 5294117647058823
:588235294117647 &times; 10 = 5882352941176470
:588235294117647 &times; 11 = 6470588235294117
:588235294117647 &times; 12 = 7058823529411764
:588235294117647 &times; 13 = 7647058823529411
:588235294117647 &times; 14 = 8235294117647058
:588235294117647 &times; 15 = 8823529411764705
:588235294117647 &times; 16 = 9411764705882352

'''乘以17出現純位數'''
:588235294117647 * 17 = 9999999999999999

长度为''L''的循环数可以表示为[[单位分数]]<math>\frac{1}{L+1}</math>小数表示形式的循环部分。反过来，如果<math>\frac{1}{p}</math>（其中''p''为[[質数]]）的循环长度为''p''-1（这样的質数''p''称为[[全循环質数]]），那么其循环部分表示的就是一个循环数。<ref>{{cite mathworld|urlname=CyclicNumber|title=Cyclic Number|access-date=2015-02-09|archive-date=2019-06-30|archive-url=https://web.archive.org/web/20190630094349/http://mathworld.wolfram.com/CyclicNumber.html|dead-url=no}}</ref>例如：
<math>{1 \over 7}=0.\overline{142857}</math>

其不同倍数的循环部分则是该循环数的循环排列：
:<math>{1 \over 7}=0.\overline{142857}</math>
:<math>{2 \over 7}=0.\overline{285714}</math>
:<math>{3 \over 7}=0.\overline{428571}</math>
:<math>{4 \over 7}=0.\overline{571428}</math>
:<math>{5 \over 7}=0.\overline{714285}</math>
:<math>{6 \over 7}=0.\overline{857142}</math>

== 參見 ==
*[[全循環質數]]

== 参考文献 ==
{{reflist}}

{{数学小作品}}

[[Category:数论]]
[[Category:置换]]