Using a parity-sensitive sieve to count prime values of a polynomial

It is expected that any irreducible polynomial with integer coefficients assumes infinitely many prime values provided that it satisfies some obvious local conditions. Moreover, it is expected that the frequency of these primes obeys a simple asymptotic law. This has however been proven for only a few special classes of polynomials. In the most famous unsolved cases the sequence of values is “thin” in the sense that it contains fewer than N(θ) integers up to N for some constant θ < 1. Quite generally it seems to be difficult to show the infinitude of primes in a given thin integer sequence and there is no polynomial for which this has hitherto been done. The polynomial x(2) + y(4) is an example of such a thin sequence; here, specifically, θ = 3/4. We report here the development of new methods that rigorously demonstrate the asymptotic formula in the case of this polynomial and that are applicable to an infinite class of polynomials to which this one belongs. The proof is based partly on a new sieve method that breaks the well-known parity problem of sieve theory and partly on a careful harmonic analysis of the special properties of biquadratic polynomial sequences.