A factoring method is presented which, heuristically, splits composite
n
n
in
O
(
n
1
/
4
+
ϵ
)
O(n^{1/4+\epsilon })
steps. There are two ideas: an integer approximation to
√
(
q
/
p
)
\surd (q/p)
provides an
O
(
n
1
/
2
+
ϵ
)
O(n^{1/2+\epsilon })
algorithm in which
n
n
is represented as the difference of two rational squares; observing that if a prime
m
m
divides a square, then
m
2
m^2
divides that square, a heuristic speed-up to
O
(
n
1
/
4
+
ϵ
)
O(n^{1/4+\epsilon })
steps is achieved. The method is well-suited for use with small computers: the storage required is negligible, and one never needs to work with numbers larger than
n
n
itself.