The Purpose of SievingEdit
Sieving at PrimeGrid is used as a quick way to minimize the number of full primality tests done. If a number has a small factor, there's no point in using a primality application on it, as that application is guaranteed to return that the number is non-prime.
Sieving does not find large prime numbers. It does find small numbers that are almost always prime (as if a candidate were found to be factored by a composite, the candidate should have already been eliminated by one of the factor's factors). But these small primes are too small to be of interest to us; they're just a means to an end.
Sieving produces a large initial payoff. The small prime 3, for instance, factors fully one in three larger numbers! However, returns diminish as the sieve continues for higher small primes. PrimeGrid policy is to terminate the sieve when it's eliminating candidates no faster than using a primality application.
Applications used at PrimeGrid for sieving: