<s>
In	O
mathematics	O
,	O
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
named	O
after	O
Belgian	O
amateur	O
mathematician	O
Edouard	O
Zeckendorf	O
,	O
is	O
a	O
theorem	O
about	O
the	O
representation	O
of	O
integers	O
as	O
sums	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
states	O
that	O
every	O
positive	O
integer	O
can	O
be	O
represented	O
uniquely	O
as	O
the	O
sum	O
of	O
one	O
or	O
more	O
distinct	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
in	O
such	O
a	O
way	O
that	O
the	O
sum	O
does	O
not	O
include	O
any	O
two	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
th	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Such	O
a	O
sum	O
is	O
called	O
the	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
of	O
.	O
</s>
<s>
The	O
Fibonacci	B-Algorithm
coding	I-Algorithm
of	O
can	O
be	O
derived	O
from	O
its	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
but	O
these	O
are	O
not	O
Zeckendorf	B-Algorithm
representations	I-Algorithm
because	O
34	O
and	O
21	O
are	O
consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
,	O
as	O
are	O
5	O
and	O
3	O
.	O
</s>
<s>
For	O
any	O
given	O
positive	O
integer	O
,	O
its	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
can	O
be	O
found	O
by	O
using	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
choosing	O
the	O
largest	O
possible	O
Fibonacci	B-Algorithm
number	I-Algorithm
at	O
each	O
stage	O
.	O
</s>
<s>
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
has	O
two	O
parts	O
:	O
</s>
<s>
Existence	O
:	O
every	O
positive	O
integer	O
has	O
a	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
Uniqueness	O
:	O
no	O
positive	O
integer	O
has	O
two	O
different	O
Zeckendorf	B-Algorithm
representations	I-Algorithm
.	O
</s>
<s>
The	O
first	O
part	O
of	O
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
(	O
existence	O
)	O
can	O
be	O
proven	O
by	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
For	O
it	O
is	O
clearly	O
true	O
(	O
as	O
these	O
are	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
)	O
,	O
for	O
we	O
have	O
.	O
</s>
<s>
If	O
is	O
a	O
Fibonacci	B-Algorithm
number	I-Algorithm
then	O
we	O
're	O
done	O
.	O
</s>
<s>
Now	O
suppose	O
each	O
positive	O
integer	O
has	O
a	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
(	O
induction	B-Algorithm
hypothesis	O
)	O
and	O
consider	O
.	O
</s>
<s>
Since	O
,	O
has	O
a	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
by	O
the	O
induction	B-Algorithm
hypothesis	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
(	O
we	O
apply	O
the	O
definition	O
of	O
Fibonacci	B-Algorithm
number	I-Algorithm
in	O
the	O
last	O
equality	O
)	O
,	O
so	O
the	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
of	O
does	O
not	O
contain	O
,	O
and	O
hence	O
also	O
does	O
not	O
contain	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
can	O
be	O
represented	O
as	O
the	O
sum	O
of	O
and	O
the	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
of	O
,	O
such	O
that	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
involved	O
in	O
the	O
sum	O
are	O
distinct	O
.	O
</s>
<s>
The	O
second	O
part	O
of	O
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
(	O
uniqueness	O
)	O
requires	O
the	O
following	O
lemma	O
:	O
</s>
<s>
Lemma	O
:	O
The	O
sum	O
of	O
any	O
non-empty	O
set	O
of	O
distinct	O
,	O
non-consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
whose	O
largest	O
member	O
is	O
is	O
strictly	O
less	O
than	O
the	O
next	O
larger	O
Fibonacci	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
The	O
lemma	O
can	O
be	O
proven	O
by	B-Algorithm
induction	I-Algorithm
on	O
.	O
</s>
<s>
Now	O
take	O
two	O
non-empty	O
sets	O
and	O
of	O
distinct	O
non-consecutive	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
which	O
have	O
the	O
same	O
sum	O
,	O
.	O
</s>
<s>
To	O
conclude	O
:	O
which	O
implies	O
,	O
proving	O
that	O
each	O
Zeckendorf	B-Algorithm
representation	I-Algorithm
is	O
unique	O
.	O
</s>
<s>
Any	O
integer	O
can	O
be	O
uniquely	O
represented	O
as	O
a	O
sum	O
of	O
negafibonacci	B-Algorithm
numbers	O
in	O
which	O
no	O
two	O
consecutive	O
negafibonacci	B-Algorithm
numbers	O
are	O
used	O
.	O
</s>
<s>
,	O
for	O
example	O
,	O
so	O
the	O
uniqueness	O
of	O
the	O
representation	O
does	O
depend	O
on	O
the	O
condition	O
that	O
no	O
two	O
consecutive	O
negafibonacci	B-Algorithm
numbers	O
are	O
used	O
.	O
</s>
<s>
This	O
gives	O
a	O
system	B-Algorithm
of	O
coding	O
integers	O
,	O
similar	O
to	O
the	O
representation	O
of	O
Zeckendorf	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
