  All sieve methods for the Goldbach problem sift out all the composite
numbers; even though, strictly speaking, it is not necessary to do so and which
is, in general, very difficult. Some new methods introduced in this paper show
that the Goldbach problem can be solved under sifting out only some composite
numbers. In fact, in order to prove the Goldbach conjecture, it is only
necessary to show that there are prime numbers left in the residual integers
after the initial sifting! This idea can be implemented by using one of the
three methods called sifting function partition by integer sort, sifting
function partition by intervals and comparative sieve method, respectively.
These are feasible methods for solving both the Goldbach problem and the
problem of twin primes. An added bonus of the above methods is the elimination
of the indeterminacy of the sifting functions brought about by their upper and
lower bounds.
