  This article develops a new sieve method which by adding an additional axiom
to the classical formulation breaks the well-known parity problem and allows
one to detect primes in thin, interesting integer sequences. In the
accompanying paper [math.NT/9811184] the practicality of the axiom is
demonstrated by verifying it (and the other axioms) to produce primes in the
sequence a^2+b^4 with the relevant asymptotics.
