Quadratic residue covers for certain real quadratic fields

Abstract:

Let ${\Delta _n}(a,b) = {(b{a^n} + (a - 1)/b)^2} + 4{a^n}$ with $n \geq 1$ and $b|a - 1$. If $\mathcal {C}$ is a finite set of primes such that for each $n \geq 1$ there exists some $q \in \mathcal {C}$ for which the Legendre symbol $({\Delta _n}(a,b)/q) \ne - 1$, we call $\mathcal {C}$ a quadratic residue cover (QRC) for the quadratic fields ${K_n}(a,b) = Q(\sqrt {{\Delta _n}(a,b))}$. It is shown how the existence of a QRC for any a, b can be used to determine lower bounds on the class number of ${K_n}(a,b)$ when ${\Delta _n}(a,b)$ is the discriminant of ${K_n}(a,b)$. Also, QRCs are computed for all $1 \leq a,b \leq 10000$.