# Eikonal Blog

## 2010.11.15

### Cyclic numbers

Filed under: mathematics, number theory — Tags: , — sandokan65 @ 14:14

Definition: (source: [3]) A number with n digits, which, when multiplied by 1, 2, 3, …, n produces the same digits in a different order. For example, 142857 is a cyclic number: 142857 × 2 = 285714; 142857 × 3 = 428571; 142857 × 4 = 571428; 142857 × 5 = 714285; 142857 × 6 = 857142, and so on. It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist.

Properties:

• $142,857 X 1 = 142,857$
• $142,857 X 2 = 285,714$
• $142,857 X 3 = 428,571$
• $142,857 X 4 = 571,428$
• $142,857 X 5 = 714,285$
• $142,857 X 6 = 857,142$
• $142,857 X 7 = 999,999$
• $142 + 857 = 999$
• $14 + 28 + 57 = 99$
• from [2]: … Multiplication by an integer greater than 7: adding the lowest six digits (ones through hundred thousands) to the remaining digits and repeat this process until you have only the six digits left, it will result in a cyclic permutation of 142857. Example:
• $142,857 X 142,857 = 20,408,122,449$
• $20,408 + 122,449 = 142,857$

Sources: