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Cyclic Number

A number having $n-1$ Digits which, when Multiplied by 1, 2, 3, ..., $n-1$, produces the same digits in a different order. Cyclic numbers are generated by the Unit Fractions $1/n$ which have maximal period Decimal Expansions (which means $n$ must be Prime). The first few numbers which generate cyclic numbers are 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (Sloane's A001913). A much larger generator is 17389.

It has been conjectured, but not yet proven, that an Infinite number of cyclic numbers exist. In fact, the Fraction of Primes which generate cyclic numbers seems to be approximately 3/8. See Yates (1973) for a table of Prime period lengths for Primes $<1,370,471$. When a cyclic number is multiplied by its generator, the result is a string of 9s. This is a special case of Midy's Theorem.










See also Decimal Expansion, Midy's Theorem


Gardner, M. Ch. 10 in Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, 1979.

Guttman, S. ``On Cyclic Numbers.'' Amer. Math. Monthly 44, 159-166, 1934.

Kraitchik, M. ``Cyclic Numbers.'' §3.7 in Mathematical Recreations. New York: W. W. Norton, pp. 75-76, 1942.

Rao, K. S. ``A Note on the Recurring Period of the Reciprocal of an Odd Number.'' Amer. Math. Monthly 62, 484-487, 1955.

Sloane, N. J. A. Sequence A001913/M4353 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Yates, S. Primes with Given Period Length. Trondheim, Norway: Universitetsforlaget, 1973.

© 1996-9 Eric W. Weisstein