<s>
Euler	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
is	O
a	O
technique	O
for	O
factoring	O
a	O
number	O
by	O
writing	O
it	O
as	O
a	O
sum	O
of	O
two	O
squares	O
in	O
two	O
different	O
ways	O
.	O
</s>
<s>
For	O
example	O
the	O
number	O
can	O
be	O
written	O
as	O
or	O
as	O
and	O
Euler	B-Language
's	O
method	O
gives	O
the	O
factorization	O
.	O
</s>
<s>
However	O
,	O
it	O
was	O
not	O
put	O
to	O
use	O
extensively	O
until	O
one	O
hundred	O
years	O
later	O
by	O
Euler	B-Language
.	O
</s>
<s>
His	O
most	O
celebrated	O
use	O
of	O
the	O
method	O
that	O
now	O
bears	O
his	O
name	O
was	O
to	O
factor	O
the	O
number	O
,	O
which	O
apparently	O
was	O
previously	O
thought	O
to	O
be	O
prime	O
even	O
though	O
it	O
is	O
not	O
a	O
pseudoprime	B-Algorithm
by	O
any	O
major	O
primality	O
test	O
.	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
is	O
more	O
effective	O
than	O
Fermat	O
's	O
for	O
integers	O
whose	O
factors	O
are	O
not	O
close	O
together	O
and	O
potentially	O
much	O
more	O
efficient	O
than	O
trial	O
division	O
if	O
one	O
can	O
find	O
representations	O
of	O
numbers	O
as	O
sums	O
of	O
two	O
squares	O
reasonably	O
easily	O
.	O
</s>
<s>
Euler	B-Language
's	O
development	O
ultimately	O
permitted	O
much	O
more	O
efficient	O
factoring	O
of	O
numbers	O
and	O
,	O
by	O
the	O
1910s	O
,	O
the	O
development	O
of	O
large	O
factor	O
tables	O
going	O
up	O
to	O
about	O
ten	O
million	O
.	O
</s>
<s>
The	O
methods	O
used	O
to	O
find	O
representations	O
of	O
numbers	O
as	O
sums	O
of	O
two	O
squares	O
are	O
essentially	O
the	O
same	O
as	O
with	O
finding	O
differences	O
of	O
squares	O
in	O
Fermat	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
great	O
disadvantage	O
of	O
Euler	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
is	O
that	O
it	O
cannot	O
be	O
applied	O
to	O
factoring	O
an	O
integer	O
with	O
any	O
prime	O
factor	O
of	O
the	O
form	O
4k+3	O
occurring	O
to	O
an	O
odd	O
power	O
in	O
its	O
prime	O
factorization	O
,	O
as	O
such	O
a	O
number	O
can	O
never	O
be	O
the	O
sum	O
of	O
two	O
squares	O
.	O
</s>
<s>
3053	O
=	O
43	O
71	O
)	O
and	O
again	O
cannot	O
be	O
factored	O
by	O
Euler	B-Language
's	O
method	O
.	O
</s>
<s>
This	O
restricted	O
applicability	O
has	O
made	O
Euler	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
disfavoured	O
for	O
computer	O
factoring	O
algorithms	O
,	O
since	O
any	O
user	O
attempting	O
to	O
factor	O
a	O
random	O
integer	O
is	O
unlikely	O
to	O
know	O
whether	O
Euler	B-Language
's	O
method	O
can	O
actually	O
be	O
applied	O
to	O
the	O
integer	O
in	O
question	O
.	O
</s>
<s>
It	O
is	O
only	O
relatively	O
recently	O
that	O
there	O
have	O
been	O
attempts	O
to	O
develop	O
Euler	B-Language
's	O
method	O
into	O
computer	O
algorithms	O
for	O
use	O
on	O
specialised	O
numbers	O
where	O
it	O
is	O
known	O
Euler	B-Language
's	O
method	O
can	O
be	O
applied	O
.	O
</s>
<s>
Euler	B-Language
's	O
method	O
relies	O
on	O
this	O
theorem	O
but	O
it	O
can	O
be	O
viewed	O
as	O
the	O
converse	O
,	O
given	O
we	O
find	O
as	O
a	O
product	O
of	O
sums	O
of	O
two	O
squares	O
.	O
</s>
