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Somewhere along the way you may have noticed that the digits in the decimal expansion of multiples of 1/7 are all rotations of the same digits:
1/7 = 0.142857142857… 2/7 = 0.285714285714… 3/7 = 0.428571428571… 4/7 = 0.571428571428… 5/7 = 0.714285714285… 6/7 = 0.857142857142…
We can make the pattern more clear by vertically aligning the sequences of digits: repeating numbers:
1/7 = 0.142857142857… 2/7 = 0.2857142857… 3/7 = 0.42857142857… 4/7 = 0.57142857… 5/7 = 0.7142857… 6/7 = 0.857142857…
Are there more cyclic fractions like that? Indeed there are. Another example is 1/17. The following shows that 1/17 how 1/7 is cyclic:
1/17 = 0.05882352941176470588235294117647… 2/17 = 0.1176470588235294117647… 3/17 = 0.176470588235294117647… 4/17 = 0.2352941176470588235294117647… 5/17 = 0.2941176470588235294117647… 6/17 = 0.352941176470588235294117647… 7/17 = 0.41176470588235294117647… 8/17 = 0.470588235294117647… 9/17 = 0.52941176470588235294117647… 10/17 = 0.5882352941176470588235294117647… 11/17 = 0.6470588235294117647… 12/17 = 0.70588235294117647… 13/17 = 0.76470588235294117647… 14/17 = 0.82352941176470588235294117647… 15/17 = 0.882352941176470588235294117647… 16/17 = 0.941176470588235294117647…
The next denominator to exhibit this pattern is 19. After finding 17 and 19 by hand, I typed “7, “7., 17, 19″ into the Online Encyclopedia of Integer Sequences found a list of denominators of cyclic fractions: OEIS A001913. These numbers are called “full reptend primes” and according to MathWorld “No general method is known for finding full reptend primes.”
