<s>
In	O
number	O
theory	O
,	O
Dixon	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
(	O
also	O
Dixon	O
's	O
random	O
squares	O
method	O
or	O
Dixon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
)	O
is	O
a	O
general-purpose	O
integer	O
factorization	O
algorithm	O
;	O
it	O
is	O
the	O
prototypical	O
factor	B-Algorithm
base	I-Algorithm
method	O
.	O
</s>
<s>
Unlike	O
for	O
other	O
factor	B-Algorithm
base	I-Algorithm
methods	O
,	O
its	O
run-time	O
bound	O
comes	O
with	O
a	O
rigorous	O
proof	O
that	O
does	O
not	O
rely	O
on	O
conjectures	O
about	O
the	O
smoothness	O
properties	O
of	O
the	O
values	O
taken	O
by	O
a	O
polynomial	O
.	O
</s>
<s>
Fermat	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
method	I-Algorithm
finds	O
such	O
a	O
congruence	O
by	O
selecting	O
random	O
or	O
pseudo-random	B-Error_Name
x	O
values	O
and	O
hoping	O
that	O
the	O
integer	O
x2modN	O
is	O
a	O
perfect	O
square	O
(	O
in	O
the	O
integers	O
)	O
:	O
</s>
<s>
If	O
there	O
are	O
many	O
numbers	O
whose	O
squares	O
can	O
be	O
factorized	O
as	O
for	O
a	O
fixed	O
set	O
of	O
small	O
primes	O
,	O
linear	B-Language
algebra	I-Language
modulo	O
2	O
on	O
the	O
matrix	O
will	O
give	O
a	O
subset	O
of	O
the	O
whose	O
squares	O
combine	O
to	O
a	O
product	O
of	O
small	O
primes	O
to	O
an	O
even	O
power	O
that	O
is	O
,	O
a	O
subset	O
of	O
the	O
whose	O
squares	O
multiply	O
to	O
the	O
square	O
of	O
a	O
(	O
hopefully	O
different	O
)	O
number	O
mod	O
N	O
.	O
</s>
<s>
Bound	O
B	O
is	O
chosen	O
,	O
and	O
the	O
factor	B-Algorithm
base	I-Algorithm
is	O
identified	O
(	O
which	O
is	O
called	O
P	O
)	O
,	O
the	O
set	O
of	O
all	O
primes	O
less	O
than	O
or	O
equal	O
to	O
B	O
.	O
</s>
<s>
When	O
enough	O
of	O
these	O
relations	O
have	O
been	O
generated	O
(	O
it	O
is	O
generally	O
sufficient	O
that	O
the	O
number	O
of	O
relations	O
be	O
a	O
few	O
more	O
than	O
the	O
size	O
of	O
P	O
)	O
,	O
the	O
methods	O
of	O
linear	B-Language
algebra	I-Language
,	O
such	O
as	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
can	O
be	O
used	O
to	O
multiply	O
together	O
these	O
various	O
relations	O
in	O
such	O
a	O
way	O
that	O
the	O
exponents	O
of	O
the	O
primes	O
on	O
the	O
right-hand	O
side	O
are	O
all	O
even	O
:	O
</s>
<s>
The	O
factor	B-Algorithm
base	I-Algorithm
is	O
then	O
P={2,3,5,7}	O
.	O
</s>
<s>
The	O
quadratic	B-Algorithm
sieve	I-Algorithm
is	O
an	O
optimization	O
of	O
Dixon	O
's	O
method	O
.	O
</s>
<s>
Also	O
,	O
the	O
size	O
of	O
the	O
factor	B-Algorithm
base	I-Algorithm
must	O
be	O
chosen	O
carefully	O
:	O
if	O
it	O
is	O
too	O
small	O
,	O
it	O
will	O
be	O
difficult	O
to	O
find	O
numbers	O
that	O
factorize	O
completely	O
over	O
it	O
,	O
and	O
if	O
it	O
is	O
too	O
large	O
,	O
more	O
relations	O
will	O
have	O
to	O
be	O
collected	O
.	O
</s>
<s>
A	O
more	O
sophisticated	O
analysis	O
,	O
using	O
the	O
approximation	O
that	O
a	O
number	O
has	O
all	O
its	O
prime	O
factors	O
less	O
than	O
with	O
probability	O
about	O
(	O
an	O
approximation	O
to	O
the	O
Dickman	O
–	O
de	O
Bruijn	O
function	O
)	O
,	O
indicates	O
that	O
choosing	O
too	O
small	O
a	O
factor	B-Algorithm
base	I-Algorithm
is	O
much	O
worse	O
than	O
too	O
large	O
,	O
and	O
that	O
the	O
ideal	O
factor	B-Algorithm
base	I-Algorithm
size	O
is	O
some	O
power	O
of	O
.	O
</s>
