Suppose we're factoring an integer via the p-1 method and we've already completed the first stage ie.

where

is the composite we wish to factor.

In the 2nd stage, we assume that there is one prime factor remaining

and go on to compute

for various prime integers.

If

is fairly smooth, would it not be more worthwhile to consider the set

for some considerably smaller integer

and then compute

for all

?

Keep in mind that we can perform another kind of "2nd stage" on this as well. ie assume that

captures most of the prime factors of

and then use a 2nd stage (3rd stage?) by computing

for various primes

and again computing

for all

.