Cyclic Number

DOWNLOAD Mathematica Notebook

A cyclic number is an (n-1)-digit integer that, when multiplied by 1, 2, 3, ..., n-1, produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913).

The decimal expansions giving the first few cyclic numbers are

1/7=0.142857^_
(1)
1/(17)=0.0588235294117647^_
(2)
1/(19)=0.052631578947368421^_
(3)
1/(23)=0.0434782608695652173913^_
(4)

(OEIS A004042).

CyclicNumberFraction

The numbers of cyclic numbers <=10^n for n=0, 1, 2, ... are 0, 1, 9, 60, 467, 3617, 25883, 248881, 2165288, 19016617, 170169241, ... (OEIS A086018). It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be Artin's constant C=0.3739558136.... The fraction of cyclic numbers among primes <=10^(10) is 0.3739551.

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 Yates (1973) for a table of prime period lengths for primes <1370471.

Wolfram Web Resources

Mathematica »

The #1 tool for creating Demonstrations and anything technical.

Wolfram|Alpha »

Explore anything with the first computational knowledge engine.

Wolfram Demonstrations Project »

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

Computerbasedmath.org »

Join the initiative for modernizing math education.

Online Integral Calculator »

Solve integrals with Wolfram|Alpha.

Step-by-step Solutions »

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.

Wolfram Problem Generator »

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.

Wolfram Education Portal »

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

Wolfram Language »

Knowledge-based programming for everyone.