On universal zero-free ternary quadratic form representations of primes in arithmetic progressions

Abstract

For the specific set of fifteen ternary quadratic forms x2+by2+cz2, b,c∈{1,2,4,8}, (b,c)∈{(2,16),(8,16),(1,3),(2,3),(1,5)}, it is shown that the distinct zero-free representations of an odd prime by these forms depend upon the class numbers h(−kp), k∈{1,3,4,5,8,12,20,24}. We determine when such a form is universal zero-free for an arithmetic progression of primes, i.e., when a prime from such a progression can be represented without zero components. The exceptional primes, which cannot be represented in this way, fall into two distinct classes. They are either infinite in number and belong to arithmetic progressions of primes, so-called infinite exceptional sets, or they are finite in number and build so-called finite exceptional sets. These exceptional sets are determined. Moreover, we show how to derive the finite number of primes expressible by a form x2+by2+cz2 in essentially m ways, and illustrate the method. Reinterpreting results by Dirichlet [12], Dickson [10] and Kaplansky [23], we show that the forms (b,c)∈{(1,2),(1,3),(2,3),(2,4)} are the only strictly universal zero-free forms of type x2+by2+cz2, i.e., they can be represented without zero components for all primes up to a known finite number of primes.