Friday, 30 June 2017

17

It's a well known fact that the proper fractions with denominator 7 are recurring decimals which share the same digits appearing cyclically:

1/7 = 0.142857
2/7 = 0.285714
3/7 = 0.428571
4/7 = 0.571428
5/7 = 0.714285
6/7 = 0.857142

It turns out that the 7ths are not the only fractions whose decimals exhibit this cyclic property.  So are the 17ths:

1/17 = 0.0588235294117647
2/17 = 0.1176470588235294
3/17 = 0.1764705882352941
4/17 = 0.2352941176470588
5/17 = 0.2941176470588235
6/17 = 0.3529411764705882
7/17 = 0.4117647058823529
8/17 = 0.4705882352941176
9/17 = 0.5294117647058823
10/17 = 0.5882352941176470
11/17 = 0.6470588235294117
12/17 = 0.7058823529411764
13/17 = 0.7647058823529411
14/17 = 0.8235294117647058
15/17 = 0.8823529411764705
16/17 = 0.9411764705882352

In fact, there are infinitely many of them: 7, 17, 19, 23, 29, 47, 59, 61, 97, ...  You must have noticed that they are all prime numbers.  They are called full reptend primes (which has 10 as their primitive roots, for those of you, like me, who are interested in algebraic number theory).

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