RATIONAL REPRESENTATIONS OF PRIMES BY BINARY QUADRATIC FORMS

Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2+ qv2has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that [Formula: see text], where [Formula: see text] is the set of norms of algebraic integers in [Formula: see text]. Let k(q) denote the smallest positive integer k such that [Formula: see text] for all q-admissible primes p. It is shown that k(q) has subexponential but suprapolynomial growth in q, as q → ∞.