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In base 10, the recurring bits of the fractions $\frac{1}7,\ldots,\frac{6}7$ are cyclic permutations of each other. e.g. $\frac{1}{7}=0.(142857)$

$\frac{2}{7}=0.(285714)$

$\frac{3}{7}=0.(428571)$

In which bases does there exist $n$ such that the recurring bits of the fractions $\frac{1}{n},\ldots,\frac{n-1}{n}$ are cyclic permutations of each other?

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Cyclic numbers exist for a base $b$ iff there is a prime $p$ such that $b$ is a primitive root mod $p$. Artin's conjecture says that there are plenty of examples. However, there are no cyclic numbers for bases that are perfect squares. See http://en.wikipedia.org/wiki/Cyclic_number.

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    @Jyrki Lahonten: Got your point. thanks for clarification.2012-01-17