cyclic permutations of periods of recurring fractions 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?

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 <