<s>
A	O
cyclic	B-Algorithm
number	I-Algorithm
is	O
an	O
integer	O
for	O
which	O
cyclic	B-Algorithm
permutations	I-Algorithm
of	O
the	O
digits	O
are	O
successive	O
integer	O
multiples	O
of	O
the	O
number	O
.	O
</s>
<s>
To	O
qualify	O
as	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
,	O
it	O
is	O
required	O
that	O
consecutive	O
multiples	O
be	O
cyclic	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
number	O
076923	O
would	O
not	O
be	O
considered	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
,	O
because	O
even	O
though	O
all	O
cyclic	B-Algorithm
permutations	I-Algorithm
are	O
multiples	O
,	O
they	O
are	O
not	O
consecutive	O
integer	O
multiples	O
:	O
</s>
<s>
repeated	O
cyclic	B-Algorithm
numbers	I-Algorithm
,	O
e.g.	O
</s>
<s>
If	O
leading	O
zeros	O
are	O
not	O
permitted	O
on	O
numerals	O
,	O
then	O
142857	O
is	O
the	O
only	O
cyclic	B-Algorithm
number	I-Algorithm
in	O
decimal	B-Application
,	O
due	O
to	O
the	O
necessary	O
structure	O
given	O
in	O
the	O
next	O
section	O
.	O
</s>
<s>
Allowing	O
leading	O
zeros	O
,	O
the	O
sequence	O
of	O
cyclic	B-Algorithm
numbers	I-Algorithm
begins	O
:	O
</s>
<s>
Cyclic	B-Algorithm
numbers	I-Algorithm
are	O
related	O
to	O
the	O
recurring	O
digital	O
representations	O
of	O
unit	O
fractions	O
.	O
</s>
<s>
then	O
the	O
digits	O
represent	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Multiples	O
of	O
these	O
fractions	O
exhibit	O
cyclic	B-Algorithm
permutation	I-Algorithm
:	O
</s>
<s>
where	O
b	O
is	O
the	O
number	O
base	O
(	O
10	O
for	O
decimal	B-Application
)	O
,	O
and	O
p	O
is	O
a	O
prime	O
that	O
does	O
not	O
divide	O
b	O
.	O
</s>
<s>
(	O
Primes	O
p	O
that	O
give	O
cyclic	B-Algorithm
numbers	I-Algorithm
in	O
base	O
b	O
are	O
called	O
full	O
reptend	O
primes	O
or	O
long	O
primes	O
in	O
base	O
b	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
case	O
b	O
=	O
10	O
,	O
p	O
=	O
7	O
gives	O
the	O
cyclic	B-Algorithm
number	I-Algorithm
142857	O
,	O
and	O
the	O
case	O
b	O
=	O
12	O
,	O
p	O
=	O
5	O
gives	O
the	O
cyclic	B-Algorithm
number	I-Algorithm
2497	O
.	O
</s>
<s>
Not	O
all	O
values	O
of	O
p	O
will	O
yield	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
using	O
this	O
formula	O
;	O
for	O
example	O
,	O
the	O
case	O
b	O
=	O
10	O
,	O
p	O
=	O
13	O
gives	O
076923076923	O
,	O
and	O
the	O
case	O
b	O
=	O
12	O
,	O
p	O
=	O
19	O
gives	O
076B45076B45076B45	O
.	O
</s>
<s>
Cyclic	B-Algorithm
numbers	I-Algorithm
can	O
be	O
constructed	O
by	O
the	O
following	O
procedure	O
:	O
</s>
<s>
if	O
t	O
=	O
p	O
−	O
1	O
then	O
n	O
is	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
This	O
procedure	O
works	O
by	O
computing	O
the	O
digits	O
of	O
1/p	O
in	O
base	O
b	O
,	O
by	O
long	B-Algorithm
division	I-Algorithm
.	O
</s>
<s>
9	O
in	O
decimal	B-Application
)	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
decimal	B-Application
,	O
142857	O
×	O
7	O
=	O
999999	O
.	O
</s>
<s>
All	O
cyclic	B-Algorithm
numbers	I-Algorithm
are	O
divisible	O
by	O
b	O
−	O
1	O
where	O
b	O
is	O
the	O
base	O
(	O
e.g.	O
</s>
<s>
9	O
in	O
decimal	B-Application
)	O
and	O
the	O
sum	O
of	O
the	O
remainder	O
is	O
a	O
multiple	O
of	O
the	O
divisor	O
.	O
</s>
<s>
Using	O
the	O
above	O
technique	O
,	O
cyclic	B-Algorithm
numbers	I-Algorithm
can	O
be	O
found	O
in	O
other	O
numeric	O
bases	O
.	O
</s>
<s>
(	O
Not	O
all	O
of	O
these	O
follow	O
the	O
second	O
rule	O
(	O
all	O
successive	O
multiples	O
being	O
cyclic	B-Algorithm
permutations	I-Algorithm
)	O
listed	O
in	O
the	O
Special	O
Cases	O
section	O
above	O
)	O
In	O
each	O
of	O
these	O
cases	O
,	O
the	O
digits	O
across	O
half	O
the	O
period	O
add	O
up	O
to	O
the	O
base	O
minus	O
one	O
.	O
</s>
<s>
Thus	O
for	O
binary	O
,	O
the	O
sum	O
of	O
the	O
bits	O
across	O
half	O
the	O
period	O
is	O
1	O
;	O
for	O
ternary	B-Algorithm
,	O
it	O
is	O
2	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
In	O
binary	O
,	O
the	O
sequence	O
of	O
cyclic	B-Algorithm
numbers	I-Algorithm
begins	O
:	O
</s>
<s>
In	O
ternary	B-Algorithm
:	O
</s>
<s>
In	O
ternary	B-Algorithm
(	O
b	O
=	O
3	O
)	O
,	O
the	O
case	O
p	O
=	O
2	O
yields	O
1	O
as	O
a	O
cyclic	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
no	O
cyclic	B-Algorithm
numbers	I-Algorithm
(	O
other	O
than	O
trivial	O
single	O
digits	O
,	O
i.e.	O
</s>
