<s>
In	O
mathematics	O
,	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
Egyptian	I-Algorithm
fractions	I-Algorithm
is	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
first	O
described	O
by	O
Fibonacci	O
,	O
for	O
transforming	O
rational	O
numbers	O
into	O
Egyptian	O
fractions	O
.	O
</s>
<s>
It	O
is	O
called	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
because	O
at	O
each	O
step	O
the	O
algorithm	O
chooses	O
greedily	O
the	O
largest	O
possible	O
unit	O
fraction	O
that	O
can	O
be	O
used	O
in	O
any	O
representation	O
of	O
the	O
remaining	O
fraction	O
.	O
</s>
<s>
He	O
includes	O
the	O
greedy	B-Algorithm
method	I-Algorithm
as	O
a	O
last	O
resort	O
for	O
situations	O
when	O
several	O
simpler	O
methods	O
fail	O
;	O
see	O
Egyptian	O
fraction	O
for	O
a	O
more	O
detailed	O
listing	O
of	O
these	O
methods	O
.	O
</s>
<s>
As	O
Salzer	O
(	O
1948	O
)	O
details	O
,	O
the	O
greedy	B-Algorithm
method	I-Algorithm
,	O
and	O
extensions	O
of	O
it	O
for	O
the	O
approximation	O
of	O
irrational	O
numbers	O
,	O
have	O
been	O
rediscovered	O
several	O
times	O
by	O
modern	O
mathematicians	O
,	O
earliest	O
and	O
most	O
notably	O
by	O
A	O
closely	O
related	O
expansion	O
method	O
that	O
produces	O
closer	O
approximations	O
at	O
each	O
step	O
by	O
allowing	O
some	O
unit	O
fractions	O
in	O
the	O
sum	O
to	O
be	O
negative	O
dates	O
back	O
to	O
.	O
</s>
<s>
The	O
expansion	O
produced	O
by	O
this	O
method	O
for	O
a	O
number	O
is	O
called	O
the	O
greedy	O
Egyptian	O
expansion	O
,	O
Sylvester	O
expansion	O
,	O
or	O
Fibonacci	B-Algorithm
–	I-Algorithm
Sylvester	I-Algorithm
expansion	I-Algorithm
of	O
.	O
</s>
<s>
However	O
,	O
the	O
term	O
Fibonacci	O
expansion	O
usually	O
refers	O
,	O
not	O
to	O
this	O
method	O
,	O
but	O
to	O
representation	O
of	O
integers	O
as	O
sums	O
of	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
The	O
greedy	B-Algorithm
method	I-Algorithm
leads	O
to	O
an	O
expansion	O
with	O
ten	O
terms	O
,	O
the	O
last	O
of	O
which	O
has	O
over	O
500	O
digits	O
in	O
its	O
denominator	O
;	O
however	O
,	O
has	O
a	O
much	O
shorter	O
non-greedy	O
representation	O
,	O
.	O
</s>
<s>
and	O
examine	O
the	O
conditions	O
under	O
which	O
the	O
greedy	B-Algorithm
method	I-Algorithm
produces	O
an	O
expansion	O
of	O
with	O
exactly	O
x	O
terms	O
;	O
these	O
can	O
be	O
described	O
in	O
terms	O
of	O
congruence	O
conditions	O
on	O
y	O
.	O
</s>
<s>
The	O
Erdős	O
–	O
Straus	O
conjecture	O
states	O
that	O
all	O
fractions	O
have	O
an	O
expansion	O
with	O
three	O
or	O
fewer	O
terms	O
,	O
but	O
when	O
such	O
expansions	O
must	O
be	O
found	O
by	O
methods	O
other	O
than	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
with	O
the	O
case	O
being	O
covered	O
by	O
the	O
congruence	O
relationship	O
.	O
</s>
<s>
and	O
describe	O
a	O
method	O
of	O
finding	O
an	O
accurate	O
approximation	O
for	O
the	O
roots	O
of	O
a	O
polynomial	O
based	O
on	O
the	O
greedy	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Some	O
,	O
though	O
not	O
labeled	O
as	O
being	O
produced	O
by	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
appear	O
to	O
be	O
of	O
the	O
same	O
type	O
.	O
</s>
