Cyclic Numbers

reptend prime
p	7

1

7

= 0.14285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714

n = cyclic number = decimal period of

1

7

= 142857

1 × n = 142857

2 × n = 285714

3 × n = 428571

4 × n = 571428

5 × n = 714285

6 × n = 857142

This Demonstration shows the first 10 cyclic numbers.

A cyclic number

n

with

p

digits is such that its digits are shifted cyclically when multiplied by an integer up to

p-1

. For example, with

n=142857

,

p=7

; multiplying by 6 shifts the digits of

x

by three places:

6×n=857142

. The decimal representation of the reciprocal of

p

has a period

n

of maximum length

p-1

. So

1/7=0.

142857

…

.

For

p>10

, leading zeros are needed for

n

. For example, the second cyclic number (which comes from

1/17=0.

0588235294117647

...

) is the integer

0588235294117647

.