Is there a sieving process that is done before the Lucas-Lehmer primality test is run to test if a Mersenne is prime?

Normally (that is for random integers), one simply uses a primordial number --- ex, product of all primes up to say 10,000, and then compute

where

is the primordial up to 10,000 and

is some integer whose primality we wish to determine.

However when it comes to Mersenne's this simple approach doesn't work too well I imagine since for a given Mersenne

, we know that any possible prime factor

is going to have to have the form

, which effectively guarantees that almost any Mersenne is going to pass this kind of weeding/screening out process since

almost all the time.

Of course I suppose if one is using an extremely large primordial like say up to

, this procedure could have a bit of merit since you've essentially got yourself a polynomial-time factoring algorithm with

as your upper limit.

As an aside, I noticed that the Lucas-Lehmer primality test can sort of be used as a bit of a Pollard-rho factoring algorithm, with

and

.

For example, if we were determining the primality of the Mersenne

, the LL-test says we need to calculate

over

where

and

. However if you stop at

and compute

you obtain the prime factor 47 and the LL-test no longer needs to continued since we know now that

is not prime.