A cyclic number is a positive integer represented by a block of decimal digits , with leading zeros retained, that,
when multiplied by 1, 2, 3, ..., cyclic permutations
of that block. Here full reptend prime (Gil and Weiner 2008),
the first few of which are 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913 ).
The decimal expansions giving the first few
cyclic numbers are
The corresponding blocks, cyclically rotated to begin with their first nonzero digit , occur in OEIS A004042 .
The numbers of full reptend primes A086018 ). Artin's conjecture predicts infinitely many cyclic
numbers and that the proportion of full reptend
primes among all primes tends to Artin's
constant full reptend primes among
primes
When a cyclic number is multiplied by its generating prime , the result is a string
of 9s. By Midy's theorem , the two equal-length halves
of its digit block also sum to a string
of 9s (Gil and Weiner 2006).
See Yates (1973) for a table of prime period lengths for primes
See also Artin's Constant ,
Decimal Expansion ,
Full Reptend Prime ,
Midy's
Theorem ,
Unique Prime
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References Gardner, M. "Cyclic Numbers." Ch. 10 in Mathematical
Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from
Scientific American. Gil,
J. B. and Weiner, M. D. "On Cyclic Numbers and an Extension of Midy's
Theorem." 12 May 2006. https://arxiv.org/abs/math/0605347 . Gil,
J. B. and Weiner, M. D. "A Quick Change of Base Algorithm for Fractions."
College Math. J. 39 , 56-59, 2008. https://www.jstor.org/stable/27646569 . Guttman,
S. "On Cyclic Numbers." Amer. Math. Monthly 41 , 159-166,
1934. https://doi.org/10.2307/2302255 . Kraitchik,
M. "Cyclic Numbers." §3.7 in Mathematical
Recreations. Numberphile.
"Cyclic Numbers." Oct. 27, 2013. https://www.youtube.com/watch?v=WUlaUalgxqI . Rao,
K. S. "A Note on the Recurring Period of the Reciprocal of an Odd Number."
Amer. Math. Monthly 62 , 484-487, 1955. Rivera, C. "Problems
& Puzzles: Puzzle 012-Period Length of https://www.primepuzzles.net/puzzles/puzz_012.htm . Sloane,
N. J. A. Sequences A001913 /M4353,
A004042 , and A086018
in "The On-Line Encyclopedia of Integer Sequences." Yates,
S. Primes with Given Period Length. Trondheim, Norway: Universitetsforlaget,
1973. Referenced on Wolfram|Alpha Cyclic Number
Cite this as:
Weisstein, Eric W. "Cyclic Number." From
MathWorld https://mathworld.wolfram.com/CyclicNumber.html
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decimal digits, with leading zeros retained, that,
when multiplied by 1, 2, 3, ..., 
, produces cyclic permutations
of that block. Here 
is a full reptend prime (Gil and Weiner 2008),
the first few of which are 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913).
for 
, 1, 2, ... are 0, 1, 9, 60, 467, 3617, 29500, 248881, 2155288,
19016617, 170169241, ... (OEIS A086018). Artin's conjecture predicts infinitely many cyclic
numbers and that the proportion of full reptend
primes among all primes tends to Artin's
constant 
.
The proportion of full reptend primes among
primes 
is approximately 0.3739552.
.
."