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Cyclic Numbers

​
reptend prime
p
7
1
7
= 0.14285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714
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
.
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