USING A PARITY-SENSITIVE SIEVE TO COUNT PRIME VALUES OF A POLYNOMIAL

It is expected that any irreducible polynomial with integer coefficien ts assumes infinitely many prime values provided that it satisfies som e obvious local conditions. Moreover, it is expected that the frequenc y of these primes obeys a simple asymptotic law. This has however been proven for only a few special classes of polynomials, In the most fam ous unsolved cases the sequence of values is ''thin'' in the sense tha t it contains fewer than N-theta integers up to N for some constant th eta < 1. Quite generally it seems to be difficult to show the infinitu de of primes in a given thin integer sequence and there is no polynomi al for which this has hitherto been done. The polynomials x(2) + y(4) is an example of such a thin sequence; here, specifically, theta = 3/4 . We report here the development of new methods that rigorously demons trate the asymptotic formula in the case of this polynomial and that a re applicable to an infinite class of polynomials to which this one be longs, 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 har monic analysis of the special properties of biquadratic polynomial seq uences.